Rational Expressions Topic 3: Adding and Subtracting Rational Expressions.
Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression...
-
Upload
moses-davidson -
Category
Documents
-
view
217 -
download
0
Transcript of Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression...
![Page 1: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/1.jpg)
Section 6.1
Rational Expressions
![Page 2: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/2.jpg)
OBJECTIVES
A Find the numbers that make a rational expression undefined.
![Page 3: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/3.jpg)
OBJECTIVES
B Write an equivalent fraction with the indicated denominator.
![Page 4: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/4.jpg)
OBJECTIVES
C Write a fraction in the standard forms.
![Page 5: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/5.jpg)
OBJECTIVES
D Reduce a fraction to lowest terms.
![Page 6: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/6.jpg)
DEFINITION
If P and Q are polynomials:
Rational Expressions
( 0)P
![Page 7: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/7.jpg)
DEFINITION
The variables in a rational expression may not be replaced by values that will make the denominator zero.
Undefined Rational Expressions
![Page 8: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/8.jpg)
DEFINITION
If P, Q, and K are polynomials
PQ
= P • KQ • K
Fundamental Property of Fractions
![Page 9: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/9.jpg)
Reducing FractionsPROCEDURE
1. Write numerator and denominator in factored form.
2. Find the GCF.
![Page 10: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/10.jpg)
Reducing FractionsPROCEDURE
3. Replace the quotient of the common factors by 1.
4. Rewrite in lowest terms.
![Page 11: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/11.jpg)
DEFINITION
Quotient of Additive Inverses
a – bb – a
= –1
![Page 12: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/12.jpg)
Practice Test
Exercise #1
Chapter 6Section 6.1A,B
![Page 13: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/13.jpg)
Find the undefined value(s) for
a. x – 2
3x + 4 is undefined when:
3x = – 4
x = – 4
3
3x + 4 = 0
![Page 14: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/14.jpg)
Write the fraction with the indicated denominator.
b. 2x 2
9y 4= ?
36y7
36y7= 9y 4 • 4y3
2x 2 • 4y3
9y 4 • 4y3 =
8x 2y3
36y 7
![Page 15: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/15.jpg)
Practice Test
Exercise #2
Chapter 6Section 6.1C
![Page 16: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/16.jpg)
Write in standard form
a. – –5
y
=
5
y
![Page 17: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/17.jpg)
Write in standard form
b. –
x – y
5
= –x + y
5
= y – x
5
![Page 18: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/18.jpg)
Practice Test
Exercise #4
Chapter 6Section 6.1D
![Page 19: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/19.jpg)
Reduce to lowest terms.
y2 – x2
x3 – y3
2 2
3 3
–1 –=
–
x y
x y
2 2
– – +=
– + +
x y x y
xy yx y x
Factor out – 1
Difference of Squares
Difference of Cubes
![Page 20: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/20.jpg)
Reduce to lowest terms.
2 2
– – +=
– + +
x y x y
xy yx y x
2 2
– +=
+ +
x y
xy yx
![Page 21: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/21.jpg)
Section 6.2
Multiplication and Division of Rational Expressions
![Page 22: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/22.jpg)
OBJECTIVES
A Multiply rational expressions.
![Page 23: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/23.jpg)
OBJECTIVES
B Divide rational expressions.
![Page 24: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/24.jpg)
OBJECTIVES
C Use multiplication and division together.
![Page 25: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/25.jpg)
DEFINITION
Multiplication of Rational Expressions
ab
• cd = a • c
b • d
( 0 0) b , d
![Page 26: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/26.jpg)
To Multiply Rational Expressions
PROCEDURE
1. Factor the numerators and denominators completely.
2. Simplify each expression.
![Page 27: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/27.jpg)
To Multiply Rational Expressions
PROCEDURE
3. Multiply remaining factors.
4. The final product should be in lowest terms.
![Page 28: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/28.jpg)
DEFINITION
Division of Real Numbers
÷a c a d =b d b c •
( and 0) b, d, c
![Page 29: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/29.jpg)
Practice Test
Exercise #6
Chapter 6Section 6.2B
![Page 30: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/30.jpg)
Perform the indicated operations.
2 – xx + 3
÷ x3 – 8x + 5
x3 + 27
x + 5
3
3– – 2 + 5 + 27
= + 3 + 5– 8
x x x
x xx
2– 2 + 2 + 4x x x
2+ 3 – 3x + 9x x
![Page 31: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/31.jpg)
2
2– – 3 + 9
= + 2 + 4
x x
x x
2
2
+ 3 – 3x + 9– – 2 + 5=
+ 3 + 5– 2 + 2 + 4
x xx xx xx x x
Perform the indicated operations.
![Page 32: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/32.jpg)
Practice Test
Exercise #7
Chapter 6Section 6.2C
![Page 33: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/33.jpg)
Perform the indicated operations.
2 – xx + 3
÷ x3 – 8x + 5
x3 + 27
x + 5
3
3– – 2 + 5 + 27
= + 3 + 5– 8
x x x
x xx
2– 2 + 2 + 4x x x
2+ 3 – 3x + 9x x
![Page 34: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/34.jpg)
2
2– – 3 + 9
= + 2 + 4
x x
x x
2
2
+ 3 – 3x + 9– – 2 + 5=
+ 3 + 5– 2 + 2 + 4
x xx xx xx x x
Perform the indicated operations.
![Page 35: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/35.jpg)
Section 6.3
Addition and Subtraction of Rational Expressions
![Page 36: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/36.jpg)
OBJECTIVES
A Add or subtract rational expressions with the same denominator.
![Page 37: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/37.jpg)
OBJECTIVES
B Add or subtract rational expressions with different denominators.
![Page 38: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/38.jpg)
Finding the LCD of Two or More Rational Expressions
PROCEDURE
1.Factor denominators. Place factors in columns.(Not necessary to factor monomials).
![Page 39: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/39.jpg)
Finding the LCD of Two or More Rational Expressions
PROCEDURE
2.Select the factor with the greatest exponent from each column.
![Page 40: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/40.jpg)
Finding the LCD of Two or More Rational Expressions
PROCEDURE
3.The product of all the factors obtained is the LCD.
![Page 41: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/41.jpg)
To Add or Subtract Fractions with Different Denominators.
PROCEDURE
1.Find the LCD.
2. Write all fractions as equivalent ones with LCDas denominator.
![Page 42: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/42.jpg)
To Add or Subtract Fractions with Different Denominators.
PROCEDURE
3.Add numerators.
4. Simplify.
![Page 43: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/43.jpg)
Practice Test
Exercise #9a
Chapter 6Section 6.3B
![Page 44: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/44.jpg)
Perform the indicated operations.
a.
x +1
x 2 + x – 2 +
x + 4
x 2 – 1
![Page 45: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/45.jpg)
Perform the indicated operations.
a.
x +1
x 2 + x – 2 +
x + 4
x 2 – 1
+1 + 4= +
+ 2 – 1 +1 – 1
x x
x x x x
= + 2 – 1 +1LCD x x x
![Page 46: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/46.jpg)
+1 + 4 = +
+ 2 – 1 +1 – 1
x x
x x x x
= + 2 – 1 +1x x x
2 2+ 2 + 1 + + 6 + 8
= + 2 – 1 +1
x x x xx x x
+ 1x
+ 1x
+ 2x
+ 2x
Perform the indicated operations.
= + 2 – 1 +1LCD x x x
2 + 2 +1x x 2+ + 6 + 8x x
![Page 47: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/47.jpg)
2 2 + + 2 + 6 +1+ 8
= + 2 – 1 +1
x x x xx x x
22 + 8 + 9
= + 2 – 1 +1
x xx x x
Perform the indicated operations.
![Page 48: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/48.jpg)
Section 6.4
Complex Fractions
![Page 49: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/49.jpg)
OBJECTIVES
A Write a complex fraction as a simple fraction in reduced form.
![Page 50: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/50.jpg)
Simplifying Complex FractionsPROCEDURE
Multiply the numerator and denominator of the complex fraction by the LCD of all simple fractions.
METHOD 1
![Page 51: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/51.jpg)
PROCEDURE
Perform operations indicated in numerator and denominator.Then divide numerator by denominator.
Simplifying Complex Fractions
METHOD 2
![Page 52: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/52.jpg)
Practice Test
Exercise #10
Chapter 6Section 6.4A
![Page 53: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/53.jpg)
Simplify.
x + 1
x2
x – 1
x3
Multiply by LCD
![Page 54: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/54.jpg)
Simplify.
2
3
3
3
1 + •
• =
1 –
xx
xx
x
x
=
x 4 + x
x 4 – 1
3
2 2
+1 =
+1 – 1
x x
x x
![Page 55: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/55.jpg)
2
2
+1 =
+1 1
x x x
x x
3
2 2
+1 =
+1 – 1
x x
x x
2
2
+1 – +1 =
+1 +1 – 1
x x x x
x x x
Simplify.
![Page 56: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/56.jpg)
Section 6.5
Division of Polynomials and Synthetic Division
![Page 57: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/57.jpg)
OBJECTIVES
A Divide a polynomial by a monomial.
![Page 58: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/58.jpg)
OBJECTIVES
B Use long division to divide one polynomial by another.
![Page 59: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/59.jpg)
OBJECTIVES
C Completely factor a polynomial when one of the factors is known.
![Page 60: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/60.jpg)
OBJECTIVES
D Use synthetic division to divide one polynomial by a binomial.
![Page 61: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/61.jpg)
OBJECTIVES
E Use the remainder theorem to verify that a number is a solution of a given equation.
![Page 62: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/62.jpg)
Dividing a Polynomial by a Monomial
RULE
Divide each term in the polynomial by the monomial.
![Page 63: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/63.jpg)
DEFINITION
The Remainder Theorem
If P(x) is divided by x –k , then the remainder is P(k).
![Page 64: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/64.jpg)
DEFINITION
The Factor Theorem
When P(x) has a factor (x –k) , it means that P(k) = 0.
![Page 65: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/65.jpg)
Practice Test
Exercise #13
Chapter 6Section 6.5B
![Page 66: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/66.jpg)
Divide.
3 2 – – 4 ÷ 2 + 26x x x
Write in descending order.
3 22 – 4 – ÷ 2+ + 20 6x x xx
Use 0x2 for missing term.
![Page 67: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/67.jpg)
– 2x – 2
– 4
2x+ 2 2x3 + 2x 2
– 2x 2
– 2x
– 2x 2 – 2x
–x –1Divide.
x 2
– 4x
– 6
Remainder
2x3 + 0x 2 – 4x – 6
![Page 68: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/68.jpg)
Practice Test
Exercise #14
Chapter 6Section 6.5C
![Page 69: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/69.jpg)
2x3 + 3x 2 – 23x – 12 x – 3
2x3 – 6x 2
9x2 – 23x
4x – 12
9x2 – 27x
Factor 2x3 + 3x2 – 23x – 12 if x – 3 is one of its factors.
2x 2
4x – 12
0
+ 9x + 4
![Page 70: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/70.jpg)
Factor 2x3 + 3x2 – 23x – 12 if x – 3 is one of its factors.
Factor 2x2 + 9x + 4
= 2x + 1 x + 4
– 3 2x + 1 x + 4 x
Factors of 2x3 + 3x2 – 23x – 12 are :
![Page 71: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/71.jpg)
Practice Test
Exercise #16
Chapter 6Section 6.5E
![Page 72: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/72.jpg)
Use synthetic division to show that – 1 is a solution of
x4 – 4x3 – 7x2 + 22x + 24 = 0
–1 1 –4 –7 22 24
–1 is a solution of the equation since the remainder R=0.
–24 +2
–1 +5
(0) 1 –5 –2 24
![Page 73: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/73.jpg)
Section 6.6
Equations Involving Rational Expressions
![Page 74: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/74.jpg)
OBJECTIVES
A Solve equations involving rational expressions.
![Page 75: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/75.jpg)
OBJECTIVES
B Solve applications using proportions.
![Page 76: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/76.jpg)
Solving Equations Containing Rational Expressions
PROCEDURE
1. Factor denominators and multiply both sides of the equation by the LCD.
![Page 77: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/77.jpg)
PROCEDURE
2. Write the result in reduced form. Use the distributive property to remove parentheses.
Solving Equations Containing Rational Expressions
![Page 78: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/78.jpg)
PROCEDURE
3. Determine whether the equation is linear or quadratic and solve accordingly.
Solving Equations Containing Rational Expressions
![Page 79: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/79.jpg)
PROCEDURE
4. Check that the proposed solution satisfies the equation. If not, discard it as an extraneous solution.
Solving Equations Containing Rational Expressions
![Page 80: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/80.jpg)
DEFINITION
Property of Proportions
If (where 0),
then
a c = b, db d
a d = b c
• •
A proportion is true if the cross products are equal.
![Page 81: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/81.jpg)
Practice Test
Exercise #18
Chapter 6Section 6.6A
![Page 82: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/82.jpg)
Solve: 18x – 2 – 3x – 1 = 1
18
x2–
3 x
= 1
x – 2 =
1
x2, x – 1 =
1x
• x2 • x2 • x2
![Page 83: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/83.jpg)
18 – 3x = x2
18
x2–
3 x
= 1 • x2 • x2 • x2
1
x
x = – 6 or x = 3
Solve: 18x – 2 – 3x – 1 = 1
0 = x2 + 3x – 18OO
0 = (x + 6)(x – 3)FF
or x – 3 = 0 x + 6 = 0FF
![Page 84: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/84.jpg)
Practice Test
Exercise #19
Chapter 6Section 6.6B
![Page 85: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/85.jpg)
A recipe for curried shrimp that normally serves four was once served to 200 guests at a wedding reception. One of the ingredients in the recipe is11
2 cups of chicken broth.
a. How much chicken broth was required to make the recipe for 200 people?
![Page 86: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/86.jpg)
People
Broth =
4
112
= 200x
4x = 200 •
32
4x = 300
x =
3004
a.
= 75 cups
![Page 87: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/87.jpg)
b. If a medium-sized can of chicken broth contains 2 cups of broth, how many cans are necessary?
75 cups
2 cups
= 37
12
38 cans
![Page 88: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/88.jpg)
Section 6.7
Applications: Problem Solving
![Page 89: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/89.jpg)
OBJECTIVES
A Solve integer problems.
![Page 90: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/90.jpg)
OBJECTIVES
B Solve work problems.
![Page 91: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/91.jpg)
OBJECTIVES
C Solve distance problems.
![Page 92: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/92.jpg)
OBJECTIVES
D Solve for a specified variable.
![Page 93: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/93.jpg)
PROCEDURE:
Read Select Think Use Verify
RSTUV Method for Solving Word Problems
![Page 94: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/94.jpg)
Practice Test
Exercise #21
Chapter 6Section 6.7B
![Page 95: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/95.jpg)
Jack can mow the lawn in 4 hours and Jill can mow it in 3. How long would it take them to mow the lawn if they work together?
let x = Time working together (hr)
Time workedTime working alone
= amount done
Jack does
x4
of the work
Jill does
x3
of the work
![Page 96: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/96.jpg)
Together they do 1 full job
x4
+ x3
= 1
3x + 4x = 12
7x = 12
Jack can mow the lawn in 4 hours and Jill can mow it in 3. How long would it take them to mow the lawn if they work together?
![Page 97: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/97.jpg)
x =
127
or 157
It takes 1
57
hours if they work together.
Jack can mow the lawn in 4 hours and Jill can mow it in 3. How long would it take them to mow the lawn if they work together?
![Page 98: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/98.jpg)
Section 6.8
Variation
![Page 99: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/99.jpg)
OBJECTIVES
A Direct variation.
![Page 100: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/100.jpg)
OBJECTIVES
B Inverse variation.
![Page 101: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/101.jpg)
OBJECTIVES
C Joint variation.
![Page 102: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/102.jpg)
OBJECTIVES
D Solve applications involving direct, inverse, and joint variation.
![Page 103: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/103.jpg)
DEFINITION
Direct Variation
y varies directly as x if there is a constant k:
y = kx
![Page 104: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/104.jpg)
DEFINITION
Inverse Variation
y varies inversely as x if there is a constant k:
y = kx
![Page 105: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/105.jpg)
DEFINITION
Joint Variation
z varies jointly with x and y if there is a constant k:
z = kxy
![Page 106: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/106.jpg)
Practice Test
Exercise #24
Chapter 6Section 6.8A
![Page 107: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/107.jpg)
C is directly proportional to m.
a. Write an equation of variation
with k as the constant.
b. Find k when C = 12 and m =
1
3 .
![Page 108: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/108.jpg)
C is directly proportional to m.
a. Write an equation of variation
with k as the constant.
C = km
Direct Variation y = kx
![Page 109: Section 6.1 Rational Expressions. OBJECTIVES A Find the numbers that make a rational expression undefined.](https://reader033.fdocuments.us/reader033/viewer/2022042703/56649eb65503460f94bbf1be/html5/thumbnails/109.jpg)
C is directly proportional to m.
C = km
12 = k
13
k = 36
b. Find k when C = 12 and m =
1
3 .