Section 2.3 Polynomial 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)

• How to use the Remainder Theorem and the Factor Theorem

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

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

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

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

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

=

2022 pp

6

2022

p

pp6

444

pp

)6(6

4464

p

ppp

4464 pp2022 pp

)(

)()(

)(

)(

xd

xrxq

xd

xf

)()()()( xrxqxdxf

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

Proper and Improper

• Since the degree of f(x) is more than or equal to d(x), the rational expression f(x)/d(x) is improper.

• Since the degree of r(x) is less than than d(x), the rational expression r(x)/d(x) is proper.

)(

)()(

)(

)(

xd

xrxq

xd

xf

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

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

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

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

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

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.

Fun with SYN and the TI-83

• Use SYN program to calculate f(-3)

• [STAT] > Edit

• Enter 1, 8, 15 into L1, then [2nd][QUIT]

• Run SYN

• Enter -3

158)( 2 xxxf )3(f

Fun with SYN and the TI-83

• Use SYN program to calculate f(-2/3)

• [STAT] > Edit

• Enter 15, 10, -6, 0, 14 into L1, then [2nd][QUIT]

• Run SYN

• Enter 2/3

1461015)( 234 xxxxf