polynomial division

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Polynomial Division Objective: To divide polynomials by long division and synthetic division

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Polynomial Division. Objective: To divide polynomials by long division and synthetic division. What you should learn. How to use long division to divide polynomials by other polynomials How to use synthetic division to divide polynomials by binomials of the form ( x – k ) - PowerPoint PPT Presentation

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Page 1: Polynomial Division

Polynomial Division

Objective:

To divide polynomials by long division and synthetic division

Page 2: Polynomial Division

What you should learn

• How to use long division to divide polynomials by other polynomials

• How to use synthetic division to divide polynomials by binomials of the form

(x – k)

• How to use the Remainder Theorem and the Factor Theorem

Page 3: Polynomial Division

641 23 xxxx

2x1. x goes into x3? x2 times.

2. Multiply (x-1) by x2.

23 xx 220 x x4

4. Bring down 4x.

5. x goes into 2x2? 2x times.

x2

6. Multiply (x-1) by 2x.

xx 22 2 x60

8. Bring down -6.

69. x goes into 6x?

6

66 x0

3. Change sign, Add.

7. Change sign, Add

6 times.

11. Change sign, Add .

10. Multiply (x-1) by 6.

3 2x x

22 2x x

6 6x

Page 4: Polynomial Division

Long Division.

1583 2 xxx

x

xx 32

155 x

5

155 x0

)5)(3( xx

Check

15352 xxx

1582 xx

2 3x x

5 15x

Page 5: Polynomial Division

Divide.

3 27

3

x

x

33 27x x

3 23 0 0 27x x x x

2x

3 23x x3 23x x 23 0x x

3x

23 9x x23 9x x 9 27x

9

9 27x 9 27x 0

Page 6: Polynomial Division

Long Division.

824 2 xxx

x

xx 42

82 x

2

82 x0

)4)(2( xx

Check

8242 xxx

822 xx

2 4x x

2 8x

Page 7: Polynomial Division

Example

2026 2 ppp

p

pp 62

204 p

4

244 p

44

6

44)6()4)(6(

pppp

Check

4424642 ppp

2022 pp

6

2022

p

pp6

44

p

2 6p p

4 24p

=

Page 8: Polynomial Division

2022 pp

6

2022

p

pp6

444

pp

)6(6

4464

p

ppp

4464 pp2022 pp

)(

)()(

)(

)(

xd

xrxq

xd

xf

)()()()( xrxqxdxf

Page 9: Polynomial Division

The Division Algorithm

If f(x) and d(x) are polynomials such that d(x) ≠ 0, and the degree of d(x) is less than or equal to the degree of f(x), there exists a unique polynomials q(x) and r(x) such that

Where r(x) = 0 or the degree of r(x) is less than the degree of d(x).

)()()()( xrxqxdxf

Page 10: Polynomial Division

Synthetic DivisionDivide x4 – 10x2 – 2x + 4 by x + 3

1 0 -10 -2 4-3

1

-3

-3

+9

-1

3

1

-3

1

3

4210 24

x

xxx

3

1

x13 23 xxx

Page 11: Polynomial Division

Long Division.

823 2 xxx

x

xx 32

8x

1

3x582)( 2 xxxf

xx 32

3 x

)3(f 8)3(2)3( 2 869

5

1 -2 -83

1

3

1

3

-5

Page 12: Polynomial Division

The Remainder Theorem

If a polynomial f(x) is divided by x – k, the remainder is r = f(k).

82)( 2 xxxf

)3(f 8)3(2)3( 2 869

5

823 2 xxx

x

xx 32

8x

1

3x5

xx 32

3 x

Page 13: Polynomial Division

The Factor TheoremA polynomial f(x) has a factor (x – k) if and only

if f(k) = 0.

Show that (x – 2) and (x + 3) are factors of

f(x) = 2x4 + 7x3 – 4x2 – 27x – 18

2 7 -4 -27 -18+2

2

4

11

22

18

36

9

18

0

Page 14: Polynomial Division

Example 6 continued

Show that (x – 2) and (x + 3) are factors of

f(x) = 2x4 + 7x3 – 4x2 – 27x – 18

2 7 -4 -27 -18+2

2

4

11

22

18

36

9

18

-3

2

-6

5

-15

3

-9

0 1827472 234 xxxx)2)(918112( 23 xxxx)3)(2)(352( 2 xxxx)3)(2)(1)(32( xxxx

Page 15: Polynomial Division

Uses of the Remainder in Synthetic Division

The remainder r, obtained in synthetic division of f(x) by (x – k), provides the following information.

1. r = f(k)

2. If r = 0 then (x – k) is a factor of f(x).

3. If r = 0 then (k, 0) is an x intercept of the graph of f.