18 April 2023 ML3 MH
Objectives : - Definition - Dividing Polynomials
Next Lesson
- Factor Theorem - Remainder Theorem
Polynomial
18 April 2023 ML3 MH
kjxcxbxax nnn ..................21
Real numbers called coefficients Constant
n is the Degree of the polynomial
Multiplying Polynomials
Expand all the terms
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126263)63)(2( 2232 xxxxxxxx
Dividing Polynomials
This is trickier than multiplication
There are two main ways
─ Long Division─ By Inspection
18 April 2023 ML3 MH
This PowerPoint presentation demonstrates two different methods of polynomial division.
Click here to see algebraic long division
Click here to see dividing “in your head”
18 April 2023 ML3 MH
Divide 2x³ + 3x² - x + 1 by x + 2
3 22 2 3 1x x x x x + 2 is the divisor
The quotient will be here.
2x³ + 3x² - x + 1 is the dividend
18 April 2023 ML3 MH
First divide the first term of the dividend, 2x³, by x (the first term of the divisor).
3 22 2 3 1x x x x
22xThis gives 2x². This will be the first term of the quotient.
18 April 2023 ML3 MH
Now multiply 2x²by x + 2
3 22 2 3 1x x x x 3 22 4x x
22x
2xand subtract
18 April 2023 ML3 MH
Bring down the next term, -x.
3 22 2 3 1x x x x 3 22 4x x
22x
2x x
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Now divide –x², the first term of –x² - x, by x, the first term of the divisor
3 22 2 3 1x x x x 3 22 4x x
22x
2x x
x
which gives –x.
18 April 2023 ML3 MH
Multiply –x by x + 2
3 22 2 3 1x x x x 3 22 4x x
22x
2x x
x
2 2x x
xand subtract
18 April 2023 ML3 MH
Bring down the next term, 1
x
3 22 2 3 1x x x x 3 22 4x x
22x
2x x
x
2 2x x
1
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Divide x, the first term of x + 1, by x, the first term of the divisor
13 22 2 3 1x x x x 3 22 4x x
22x
2x x
x
2 2x x
x 1which gives 1
18 April 2023 ML3 MH
Multiply x + 2 by 1
3 22 2 3 1x x x x 3 22 4x x
22x
2x x
x
2 2x x
x
1
12x 1and subtract
18 April 2023 ML3 MH
The remainder is –1.
3 22 2 3 1x x x x 3 22 4x x
22x
2x x
x
2 2x x
x
1
12x 1
The quotient is 2x² - x + 1
18 April 2023 ML3 MH
Click here to see this example of algebraic long division again
Click here to see dividing “in your head”
Click here to end the presentation
18 April 2023 ML3 MH
Divide 2x³ + 3x² - x + 1 by x + 2
When a cubic is divided by a linear expression, the quotient is a quadratic and the remainder, if any, is a constant.
Let the remainder be d.
Let the quotient by ax² + bx + c
2x³ + 3x² - x + 1 = (x + 2)(ax² + bx + c) + d
18 April 2023 ML3 MH
2x³ + 3x² - x + 1 = (x + 2)(ax² + bx + c) + d
The first terms in each bracket give the term in x³
x multiplied by ax² gives ax³
so a must be 2.
18 April 2023 ML3 MH
2x³ + 3x² - x + 1 = (x + 2)(2x² + bx + c) + d
The first terms in each bracket give the term in x³
x multiplied by ax² gives ax³
so a must be 2.
18 April 2023 ML3 MH
2x³ + 3x² - x + 1 = (x + 2)(2x² + bx + c) + d
Now look for pairs of terms that multiply to give terms in x²
x multiplied by bx gives bx²
bx² + 4x² must be 3x²
2 multiplied by 2x² gives 4x²
so b must be -1.
18 April 2023 ML3 MH
2x³ + 3x² - x + 1 = (x + 2)(2x² + -1x + c) + d
Now look for pairs of terms that multiply to give terms in x²
x multiplied by bx gives bx²
bx² + 4x² must be 3x²
2 multiplied by 2x² gives 4x²
so b must be -1.
18 April 2023 ML3 MH
2x³ + 3x² - x + 1 = (x + 2)(2x² - x + c) + d
Now look for pairs of terms that multiply to give terms in x
x multiplied by c gives cx
cx - 2x must be -x
2 multiplied by -x gives -2x
so c must be 1.
18 April 2023 ML3 MH
2x³ + 3x² - x + 1 = (x + 2)(2x² - x + 1) + d
Now look for pairs of terms that multiply to give terms in x
x multiplied by c gives cx
cx - 2x must be -x
2 multiplied by -x gives -2x
so c must be 1.
18 April 2023 ML3 MH
2x³ + 3x² - x + 1 = (x + 2)(2x² - x + 1) + d
Now look at the constant term
2 multiplied by 1 gives 2
2 + d must be 1
then add d
so d must be -1.
18 April 2023 ML3 MH
2x³ + 3x² - x + 1 = (x + 2)(2x² - x + 1) - 1
Now look at the constant term
2 multiplied by 1 gives 2
2 + d must be 1
then add d
so d must be -1.
18 April 2023 ML3 MH
2x³ + 3x² - x + 1 = (x + 2)(2x² - x + 1) - 1
The quotient is 2x² - x + 1 and the remainder is –1.
18 April 2023 ML3 MH
Click here to see algebraic long division
Click here to see this example of dividing “in your head” again
Click here to end the presentation
18 April 2023 ML3 MH
Do the following
1.
2.
3.
4.
18 April 2023 ML3 MH
)12()7136( 23 xxxx
)3()15171392( 234 xxxxx
)47()21643283( 2234 xxxxxx
)1()23( 23 xxx
Exercises C1/C2 Page 82 Ex 3A, Nos 3, 6, 9, 16 to 20
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