13 multiplication and division of rational expressions
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Transcript of 13 multiplication and division of rational expressions
![Page 1: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/1.jpg)
Multiplication and Division of Rational Expressions
![Page 2: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/2.jpg)
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
![Page 3: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/3.jpg)
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.
![Page 4: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/4.jpg)
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.
Example A. Simplify
10xy3z
a. * y2
5x3
![Page 5: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/5.jpg)
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.
Example A. Simplify
10xy3z
a. * y2
5x3 =
10xy2
5x3y3z
![Page 6: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/6.jpg)
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.
Example A. Simplify
10xy3z
a. * y2
5x3 =
10xy2
5x3y3z =
2x2yz
![Page 7: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/7.jpg)
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.
Example A. Simplify
10xy3z
a. * y2
5x3 =
10xy2
5x3y3z =
2x2yz
b. (x2 + 2x – 3 ) (x – 2) (x2 – x )
(x2 – 4 )*
![Page 8: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/8.jpg)
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.
Example A. Simplify
10xy3z
a. * y2
5x3 =
10xy2
5x3y3z =
2x2yz
b. (x2 + 2x – 3 ) (x – 2) (x2 – x )
(x2 – 4 )*
=
(x + 3)(x – 1 )
![Page 9: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/9.jpg)
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.
Example A. Simplify
10xy3z
a. * y2
5x3 =
10xy2
5x3y3z =
2x2yz
b. (x2 + 2x – 3 ) (x – 2) (x2 – x )
(x2 – 4 )*
=
(x + 3)(x – 1 ) (x + 2 )(x – 2)
![Page 10: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/10.jpg)
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.
Example A. Simplify
10xy3z
a. * y2
5x3 =
10xy2
5x3y3z =
2x2yz
b. (x2 + 2x – 3 ) (x – 2) (x2 – x )
(x2 – 4 )*
=
(x + 3)(x – 1 ) (x – 2)
(x + 2 )(x – 2)
![Page 11: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/11.jpg)
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.
Example A. Simplify
10xy3z
a. * y2
5x3 =
10xy2
5x3y3z =
2x2yz
b. (x2 + 2x – 3 ) (x – 2) (x2 – x )
(x2 – 4 )*
=
(x + 3)(x – 1 ) (x – 2) x(x – 1 )
(x + 2 )(x – 2)
![Page 12: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/12.jpg)
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.
Example A. Simplify
10xy3z
a. * y2
5x3 =
10xy2
5x3y3z =
2x2yz
b. (x2 + 2x – 3 ) (x – 2) (x2 – x )
(x2 – 4 )*
=
(x + 3)(x – 1 ) (x – 2) x(x – 1 )
(x + 2 )(x – 2)
![Page 13: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/13.jpg)
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.
Example A. Simplify
10xy3z
a. * y2
5x3 =
10xy2
5x3y3z =
2x2yz
b. (x2 + 2x – 3 ) (x – 2) (x2 – x )
(x2 – 4 )*
=
(x + 3)(x – 1 ) (x – 2) x(x – 1 )
(x + 2 )(x – 2)
![Page 14: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/14.jpg)
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.
Example A. Simplify
10xy3z
a. * y2
5x3 =
10xy2
5x3y3z =
2x2yz
b. (x2 + 2x – 3 ) (x – 2) (x2 – x )
(x2 – 4 )
=(x + 3)(x + 2)
x
*
=
(x + 3)(x – 1 ) (x – 2) x(x – 1 )
(x + 2 )(x – 2)
In the next section, we meet the following type of problems.
![Page 15: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/15.jpg)
Multiplication and Division of Rational ExpressionsExample B. Simplify and expand the answers.
a. x + 3 x – 1 (x2 – 1)
![Page 16: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/16.jpg)
Multiplication and Division of Rational Expressions
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
Example B. Simplify and expand the answers.
![Page 17: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/17.jpg)
Multiplication and Division of Rational ExpressionsExample B. Simplify and expand the answers.
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1)
![Page 18: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/18.jpg)
Multiplication and Division of Rational Expressions
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1) = x2 + 4x + 3
Example B. Simplify and expand the answers.
![Page 19: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/19.jpg)
Multiplication and Division of Rational Expressions
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1) = x2 + 4x + 3
b. x – 2 x2 – 9 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)
Example B. Simplify and expand the answers.
![Page 20: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/20.jpg)
Multiplication and Division of Rational Expressions
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1) = x2 + 4x + 3
b. x – 2 x2 – 9 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)
=
x – 2 (x – 3)(x + 3)
Example B. Simplify and expand the answers.
![Page 21: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/21.jpg)
Multiplication and Division of Rational Expressions
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1) = x2 + 4x + 3
b. x – 2 x2 – 9 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)
=
x – 2 (x – 3)(x + 3) – x + 1
(x – 3)(x + 1)
Example B. Simplify and expand the answers.
![Page 22: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/22.jpg)
Multiplication and Division of Rational Expressions
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1) = x2 + 4x + 3
b. x – 2 x2 – 9 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)
=
x – 2 (x – 3)(x + 3) [ – x + 1
(x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1)
Example B. Simplify and expand the answers.
![Page 23: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/23.jpg)
Multiplication and Division of Rational Expressions
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1) = x2 + 4x + 3
b. x – 2 x2 – 9 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)
=
x – 2 (x – 3)(x + 3) [ – x + 1
(x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1)
Example B. Simplify and expand the answers.
![Page 24: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/24.jpg)
Multiplication and Division of Rational Expressions
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1) = x2 + 4x + 3
b. x – 2 x2 – 9 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)
=
x – 2 (x – 3)(x + 3) [ – x + 1
(x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3)
Example B. Simplify and expand the answers.
![Page 25: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/25.jpg)
Multiplication and Division of Rational Expressions
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1) = x2 + 4x + 3
b. x – 2 x2 – 9 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)
=
x – 2 (x – 3)(x + 3) [ – x + 1
(x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3)
=
(x – 2)(x + 1) – (x + 1)(x + 3)
Example B. Simplify and expand the answers.
![Page 26: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/26.jpg)
Multiplication and Division of Rational Expressions
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1) = x2 + 4x + 3
b. x – 2 x2 – 9 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)
=
x – 2 (x – 3)(x + 3) [ – x + 1
(x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3)
=
(x – 2)(x + 1) – (x + 1)(x + 3)
= (x – 2)(x + 1) + (–x –1)(x + 3)
Example B. Simplify and expand the answers.
![Page 27: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/27.jpg)
Multiplication and Division of Rational Expressions
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1) = x2 + 4x + 3
b. x – 2 x2 – 9 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)
=
x – 2 (x – 3)(x + 3) [ – x + 1
(x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3)
=
(x – 2)(x + 1) – (x + 1)(x + 3)
= (x – 2)(x + 1) + (–x –1)(x + 3)
= x2 – x – 2 – x2 – 4x – 3
Example B. Simplify and expand the answers.
![Page 28: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/28.jpg)
Multiplication and Division of Rational ExpressionsExample B. Simplify and expand the answers.
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1) = x2 + 4x + 3
b. x – 2 x2 – 9 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)
=
x – 2 (x – 3)(x + 3) [ – x + 1
(x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3)
=
(x – 2)(x + 1) – (x + 1)(x + 3)
= (x – 2)(x + 1) + (–x –1)(x + 3)
= x2 – x – 2 – x2 – 4x – 3
= –5x – 5
![Page 29: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/29.jpg)
Division Rule for Rational Expressions
Multiplication and Division of Rational Expressions
AB
CD
÷
![Page 30: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/30.jpg)
Division Rule for Rational Expressions
Multiplication and Division of Rational Expressions
AB
CD
÷ = ADBC
Reciprocate
![Page 31: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/31.jpg)
Division Rule for Rational Expressions
Multiplication and Division of Rational Expressions
AB
CD
÷ = ADBC
Reciprocate
We convert division by an expression of multiplying by its reciprocal.
![Page 32: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/32.jpg)
Division Rule for Rational Expressions
Multiplication and Division of Rational Expressions
AB
CD
÷ = ADBC
Reciprocate
We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
![Page 33: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/33.jpg)
Division Rule for Rational Expressions
Multiplication and Division of Rational Expressions
AB
CD
÷ = ADBC
Reciprocate
(2x – 6) (x + 3)
÷(x2 + 2x – 3)
(9 – x2)
Example C. Simplify
We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
![Page 34: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/34.jpg)
Division Rule for Rational Expressions
Multiplication and Division of Rational Expressions
AB
CD
÷ = ADBC
Reciprocate
(2x – 6) (x + 3)
÷(x2 + 2x – 3)
(9 – x2)
=(2x – 6) (x + 3)
(x2 + 2x – 3) (9 – x2)
*
Example C. Simplify
We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
![Page 35: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/35.jpg)
Division Rule for Rational Expressions
Multiplication and Division of Rational Expressions
AB
CD
÷ = ADBC
Reciprocate
(2x – 6) (x + 3)
÷(x2 + 2x – 3)
(9 – x2)
=(2x – 6) (x + 3)
(x2 + 2x – 3) (9 – x2)
*
=2(x – 3) (x + 3)
Example C. Simplify
We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
![Page 36: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/36.jpg)
Division Rule for Rational Expressions
Multiplication and Division of Rational Expressions
AB
CD
÷ = ADBC
Reciprocate
(2x – 6) (x + 3)
÷(x2 + 2x – 3)
(9 – x2)
=(2x – 6) (x + 3)
(x2 + 2x – 3) (9 – x2)
*
=2(x – 3) (x + 3)
(x + 3)(x – 1) (3 – x)(3 + x)
Example C. Simplify
We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
![Page 37: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/37.jpg)
Division Rule for Rational Expressions
Multiplication and Division of Rational Expressions
AB
CD
÷ = ADBC
Reciprocate
Example C. Simplify
(2x – 6) (x + 3)
÷(x2 + 2x – 3)
(9 – x2)
=(2x – 6) (x + 3)
(x2 + 2x – 3) (9 – x2)
*
=2(x – 3) (x + 3)
(x + 3)(x – 1) (3 – x)(3 + x)
*
(9 – x2)
We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
![Page 38: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/38.jpg)
Division Rule for Rational Expressions
Multiplication and Division of Rational Expressions
AB
CD
÷ = ADBC
Reciprocate
(2x – 6) (x + 3)
÷(x2 + 2x – 3)
(9 – x2)
=(2x – 6) (x + 3)
(x2 + 2x – 3) (9 – x2)
*
=2(x – 3) (x + 3)
(x + 3)(x – 1) (3 – x)(3 + x)
*
(–1)
Example C. Simplify
We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
![Page 39: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/39.jpg)
Division Rule for Rational Expressions
Multiplication and Division of Rational Expressions
AB
CD
÷ = ADBC
Reciprocate
(2x – 6) (x + 3)
÷(x2 + 2x – 3)
(9 – x2)
=(2x – 6) (x + 3)
(x2 + 2x – 3) (9 – x2)
*
=2(x – 3) (x + 3)
(x + 3)(x – 1) (3 – x)(3 + x)
*
(–1)
=–2(x – 1)
(3 + x)
Example C. Simplify
We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
![Page 40: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/40.jpg)
Multiplication and Division of Rational ExpressionsBesides the expanded form and factored forms, rational expressions may also be split into sums or differences.
![Page 41: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/41.jpg)
Multiplication and Division of Rational ExpressionsBesides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this.
![Page 42: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/42.jpg)
Multiplication and Division of Rational ExpressionsBesides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term.
![Page 43: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/43.jpg)
Multiplication and Division of Rational Expressions
Example D. Break up the numerators as the sums or differences and simplify each term.
(2x – 6) (x + 3)
a. =
Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term.
(2x – 6) 3x2
b. =
![Page 44: 13 multiplication and division of rational expressions](https://reader035.fdocuments.us/reader035/viewer/2022062515/55cfa137bb61eb735c8b4600/html5/thumbnails/44.jpg)
Multiplication and Division of Rational Expressions
Example D. Break up the numerators as the sums or differences and simplify each term.
(2x – 6) (x + 3)
a. =
Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term.
(x + 3) – (x + 3)2x 6
(2x – 6) 3x2
b. =
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Multiplication and Division of Rational Expressions
Example D. Break up the numerators as the sums or differences and simplify each term.
(2x – 6) (x + 3)
a. =
Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term.
(x + 3) – (x + 3)2x 6
(2x – 6) 3x2
b. = – 2x 63x2 3x2
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Multiplication and Division of Rational Expressions
Example D. Break up the numerators as the sums or differences and simplify each term.
(2x – 6) (x + 3)
a. =
Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term.
(x + 3) – (x + 3)2x 6
(2x – 6) 3x2
b. = – 2x 63x2 3x2 = – 2
x22
3x
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Multiplication and Division of Rational Expressions
Example D. Break up the numerators as the sums or differences and simplify each term.
(2x – 6) (x + 3)
a. =
Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term.
(x + 3) – (x + 3)2x 6
(2x – 6) 3x2
b. = – 2x 63x2 3x2 = – 2
x22
3xII. Long DivisionLong division is the extension of the long division of numbers from grade school and it is for the division of polynomials in one variable.
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Multiplication and Division of Rational Expressions
Example D. Break up the numerators as the sums or differences and simplify each term.
(2x – 6) (x + 3)
a. =
Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term.
(x + 3) – (x + 3)2x 6
(2x – 6) 3x2
b. = – 2x 63x2 3x2 = – 2
x22
3xII. Long DivisionLong division is the extension of the long division of numbers from grade school and it is for the division of polynomials in one variable. Specifically, long division gives relevant results only when the degree of the numerator is the same or more than the degree of the denominator.
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Multiplication and Division of Rational ExpressionsLet’s look at the example 125/8 or 125 ÷ 8 by long division.
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Multiplication and Division of Rational ExpressionsLet’s look at the example 125/8 or 125 ÷ 8 by long division.i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient.
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Multiplication and Division of Rational ExpressionsLet’s look at the example 125/8 or 125 ÷ 8 by long division.i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125
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Multiplication and Division of Rational ExpressionsLet’s look at the example 125/8 or 125 ÷ 8 by long division.i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125
1
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Multiplication and Division of Rational ExpressionsLet’s look at the example 125/8 or 125 ÷ 8 by long division.i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125
1
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
845
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Multiplication and Division of Rational ExpressionsLet’s look at the example 125/8 or 125 ÷ 8 by long division.i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125
1
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
845
iii. Repeat steps i and ii until no more quotient may be entered.
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Multiplication and Division of Rational Expressions
i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125
15
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
845
iii. Repeat steps i and ii until no more quotient may be entered.
405
Let’s look at the example 125/8 or 125 ÷ 8 by long division.
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Multiplication and Division of Rational Expressions
i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125
15
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
845
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:ND = Q + R
Dand that R (the remainder) is smaller then D (no more quotient).
405
where Q is the quotient
Let’s look at the example 125/8 or 125 ÷ 8 by long division.
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Multiplication and Division of Rational Expressions
i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125
15
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
845
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:ND = Q + R
Dand that R (the remainder) is smaller then D (no more quotient).
405
1258 = 15 + 5
8
where Q is the quotient
Let’s look at the example 125/8 or 125 ÷ 8 by long division.
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Multiplication and Division of Rational ExpressionsExample E. Divide using long division(2x – 6)
(x + 3)
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Multiplication and Division of Rational Expressions
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example E. Divide using long division(2x – 6) (x + 3)
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Multiplication and Division of Rational Expressions
)x + 3 2x – 6
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Make sure the terms are in order.
Example E. Divide using long division(2x – 6) (x + 3)
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Multiplication and Division of Rational Expressions
)x + 3 2x – 6
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Make sure the terms are in order.
Example E. Divide using long division(2x – 6) (x + 3)
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Multiplication and Division of Rational Expressions
)x + 3 2x – 6
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
enter the quotients of the leading terms 2x/x = 2
Example E. Divide using long division(2x – 6) (x + 3)
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Multiplication and Division of Rational Expressions
)x + 3 2x – 6 2
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
enter the quotients of the leading terms 2x/x = 2
Example E. Divide using long division(2x – 6) (x + 3)
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Multiplication and Division of Rational Expressions
)x + 3 2x – 6 2
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example E. Divide using long division(2x – 6) (x + 3)
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Multiplication and Division of Rational Expressions
)x + 3 2x – 6 2
2x + 6ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example E. Divide using long division(2x – 6) (x + 3)
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Multiplication and Division of Rational Expressions
)x + 3 2x – 6 2
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
2x + 6–12
–)
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example E. Divide using long division(2x – 6) (x + 3)
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Multiplication and Division of Rational Expressions
)x + 3 2x – 6 2
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
2x + 6–12
iii. Repeat steps i and ii until no more quotient may be entered.
–)
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example E. Divide using long division(2x – 6) (x + 3)
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Multiplication and Division of Rational Expressions
)x + 3 2x – 6 2
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
2x + 6–12
–)
Stop. No more quotient since x can’t going into 12.iii. Repeat steps i and ii until no more
quotient may be entered.
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example E. Divide using long division(2x – 6) (x + 3)
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Multiplication and Division of Rational Expressions
)x + 3 2x – 6 2
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
2x + 6–12
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
–)
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example E. Divide using long division(2x – 6) (x + 3)
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Multiplication and Division of Rational Expressions
)x + 3 2x – 6 2
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
2x + 6–12
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
–)
Hence we may write(2x – 6) (x + 3)
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example E. Divide using long division(2x – 6) (x + 3)
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Multiplication and Division of Rational Expressions
)x + 3 2x – 6 2
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
2x + 6–12
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
–)
Hence we may write(2x – 6) (x + 3)
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Q
R
Example E. Divide using long division(2x – 6) (x + 3)
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Multiplication and Division of Rational Expressions
)x + 3 2x – 6 2
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
2x + 6–12
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
= 2 – 12 x + 3
–)
Hence we may write(2x – 6) (x + 3)
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Q
R
Q R
Example E. Divide using long division(2x – 6) (x + 3)
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Multiplication and Division of Rational ExpressionsExample F. Divide using long division
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
x2 – 6x + 3x – 2
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
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Multiplication and Division of Rational Expressions
)x + 3
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
x2 – 6x + 3
Make sure the terms are in order.
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example F. Divide using long divisionx2 – 6x + 3x – 2
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Multiplication and Division of Rational Expressions
)x + 3
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
x2 – 6x + 3
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example F. Divide using long divisionx2 – 6x + 3x – 2
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Multiplication and Division of Rational Expressions
)x + 3 x
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
x2 – 6x + 3
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example F. Divide using long divisionx2 – 6x + 3x – 2
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Multiplication and Division of Rational Expressions
)x + 3 x
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
x2 + 3x
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
x2 – 6x + 3
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example F. Divide using long divisionx2 – 6x + 3x – 2
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Multiplication and Division of Rational Expressions
)x + 3 x
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
x2 + 3x–9x + 3
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
–)x2 – 6x + 3
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example F. Divide using long divisionx2 – 6x + 3x – 2
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Multiplication and Division of Rational Expressions
)x + 3 x – 9
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
x2 + 3x–9x + 3
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
–)x2 – 6x + 3
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example F. Divide using long divisionx2 – 6x + 3x – 2
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Multiplication and Division of Rational Expressions
)x + 3 x – 9
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
x2 + 3x–9x + 3
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
–)x2 – 6x + 3
–9x – 27
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example F. Divide using long divisionx2 – 6x + 3x – 2
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Multiplication and Division of Rational Expressions
)x + 3 x – 9
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
x2 + 3x–9x + 3
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
–)x2 – 6x + 3
–9x – 27–)30
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example F. Divide using long divisionx2 – 6x + 3x – 2
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Multiplication and Division of Rational Expressions
)x + 3 x – 9
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
x2 + 3x–9x + 3
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient
–)x2 – 6x + 3
–9x – 27–)30
Stop. No more quotient since x can’t going into 30. Hence 30 is the remainder.
and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example F. Divide using long divisionx2 – 6x + 3x – 2
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Multiplication and Division of Rational Expressions
)x + 3 x – 9
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
x2 + 3x–9x + 3
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
–)x2 – 6x + 3
–9x – 27–)30
Hence
x2 – 6x + 3x – 2
=
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example F. Divide using long divisionx2 – 6x + 3x – 2
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Multiplication and Division of Rational Expressions
)x + 3 x – 9
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
x2 + 3x–9x + 3
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
–)x2 – 6x + 3
–9x – 27–)30
Hence
x2 – 6x + 3x – 2
= x – 9 + 30x + 3
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
Example F. Divide using long divisionx2 – 6x + 3x – 2
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Ex A. Simplify. Do not expand the results. Multiplication and Division of Rational Expressions
1. 10x * 25x3
15x4
* 1625x4
10x* 35x32.
5. 109x4
* 185x3
6.
3.12x6* 5
6x14
56x6
27* 63
8x5
10x* 35x34.
7. 75x49
* 4225x3
8.
9.2x – 4 2x + 4
5x + 10 3x – 6
10.6 – 4x 3x – 2
x – 2 2x + 4
11.9x – 12 2x – 4
2 – x 8 – 6x
12. x + 4
–x – 44 – x
x – 4
13.3x – 9
15x – 53 – x
5 – 15x14.
42 – 6x –2x + 14
4 – 2x –7x + 14
*
*
*
*
*
*
15.(x2 + x – 2 ) (x – 2) (x2 – x)
(x2 – 4 )*
16. (x2 + 2x – 3 ) (x2 – 9) (x2 – x – 2 )
(x2 – 2x – 3)*
17.(x2 – x – 2 ) (x2 – 1) (x2 + 2x + 1)
(x2 + x )*
18. (x2 + 5x – 6 ) (x2 + 5x + 6) (x2 – 5x – 6 )
(x2 – 5x + 6)*
19.(x2 – 3x – 4 ) (x2 – 1) (x2 – 2x – 8)
(x2 – 3x + 2)*
20.(– x2 + 6 – x ) (x2 + 5x + 6) (x2 – x – 12 )
(6 – x2 – x)*
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Ex. A. Simplify. Do not expand the results. Multiplication and Division of Rational Expressions
21. (2x2 + x – 1 ) (1 – 2x)
(4x2 – 1) (2x2 – x )
22. (3x2 – 2x – 1) (1 – 9x2)
(x2 + x – 2 ) (x2 + 4x + 4)
23.(3x2 – x – 2) (x2 – x + 2) (3x2 + 4x + 1)
(–x – 3x2)24. (x + 1 – 6x2)
(–x2 – 4)(2x2 + x – 1 ) (x2 – 5x – 6)
25. (x3 – 4x) (–x2 + 4x – 4)
(x2 + 2) (–x + 2)
26. (–x3 + 9x ) (x2 + 6x + 9)(x2 + 3x) (–3x2 – 9x)
Ex. B. Multiply, expand and simplify the results.
÷
÷
÷
÷
÷
÷
27. x + 3 x + 1 (x2 – 1) 28. x – 3
x – 2 (x2 – 4) 29. 2x + 3 1 – x (x2 – 1)
30. 3 – 2x x + 2 (x + 2)(x +1) 31. 3 – 2x
2x – 1 (3x + 2)(1 – 2x)
32. x – 2 x – 3 ( + x + 1
x + 3 )( x – 3)(x + 3)
33. 2x – 1 x + 2 ( – x + 2
2x – 3 ) ( 2x – 3)(x + 2)
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Multiplication and Division of Rational Expressions
38. x – 2 x2 – 9 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)
39. x + 3 x2 – 4 ( – 2x + 1
x2 + x – 2 ) ( x – 2)(x + 2)(x – 1)
40. x – 1 x2 – x – 6 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 2)(x + 1)
41. x + 2 x2 – 4x +3( – 2x + 1
x2 + 2x – 3 ) ( x – 3)(x + 3)(x – 1)
34. 4 – x x – 3 ( – x – 1
2x + 3 )( x – 3)(2x + 3)
35. 3 – x x + 2 ( – 2x + 3
x – 3 )(x – 3)(x + 2)
Ex B. Multiply, expand and simplify the results.
36. 3 – 4x x + 1 ( – 1 – 2x
x + 3 )( x + 3)(x + 1)
37. 5x – 7 x + 5 ( – 4 – 5x
x – 3 )(x – 3)(x + 5)
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Ex. C. Break up the following expressions as sums and differences of fractions.
42. 43. 44.
45. 46. 47.
x2 + 4x – 6 2x2x2 – 4
x2
12x3 – 9x2 + 6x3x
x2 – 4 2x
xx8 – x6 – x4
x2x8 – x6 – x4
Ex D. Use long division and write each rational expression in
the form of Q + .RD
(x2 + x – 2 ) (x – 1)
(3x2 – 3x – 2 ) (x + 2)
2x + 6 x + 2 48. 3x – 5
x – 2 49. 4x + 3 x – 1 50.
5x – 4 x – 3 51. 3x + 8
2 – x52. –4x – 5 1 – x53.
54. (2x2 + x – 3 ) (x – 2)
55. 56.
(–x2 + 4x – 3 ) (x – 3)
(5x2 – 1 ) (x – 4)
57. (4x2 + 2 ) (x + 3)
58. 59.
Multiplication and Division of Rational Expressions