ACTIVITY 31:

Dividing Polynomials (Section 4.2, pp. 325-331)

Example 1:

Divide 63 by 12.

rq 12*63 122436486072

312*563

Division Algorithm:

If P(x) and D(x) are polynomials, with D(x) ≠ 0, then there exist unique polynomials Q(x) and R(x), where R(x) is either 0 or of degree strictly less than the degree of D(x), such that

P(x) = Q(x)D(x) + R(x)The polynomials P(x) and D(x) are called the dividend and divisor, respectively; Q(x) is the quotientand R(x) is the remainder.

Example 2:

Divide the polynomial P(x) = 2x2 − x − 4 by D(x) = x − 3.

2x2 − x − 4 x − 3

2x

2x2 – 6x-2x2 + 6x

5x – 4

+ 5

5x – 15 -5x + 15

11

)(xQ

)(xR

)()()()( xRxDxQxP 1135242 2 xxxx

Example 3:

x4 − x3 + 4x + 2x2 + 3

x2

x4 + 3x2 –x4 – 3x2

− x3 – 3x2 + 4x + 2

– x

– x3 – 3x + x3 + 3x

– 3x2 + 7x + 2

)(xQ

)(xR

)()()()( xRxDxQxP 1173324 2234 xxxxxxx

Divide the polynomial P(x) = x4 − x3 + 4x + 2 by D(x) = x2 + 3.

– 3

– 3x2 – 9 +3x2 + 9

7x + 11

Synthetic Division:

Use synthetic division to divide the polynomialP(x) = 2x2 − x − 4 by D(x) = x − 3.

root 23 1 4

2

6

5

15

11

)(xQ )(xR

)()()()( xRxDxQxP 1135242 2 xxxx

52)( xxQ

Example 4:

Use synthetic division to find the quotient Q(x) and the remainder R(x) when:

f(x) = 3x3 + 2x2 − x + 3 is divided by g(x) = x − 4.

34 2 1

3

12

14

56

55

)(xQ )(xR55143)( 2 xxxQ

3

220

223

223)( xR

Example 5:

Use synthetic division to find the quotient Q(x) and the remainder R(x) when:

f(x) = x5 − 4x3 + x is divided by g(x) = x + 3.

13 0 4

1

3

3

9

5

)(xQ )(xR461553)( 234 xxxxxQ

0

15

15

138)( xR

1 0

45

46

138

138

Remainder Theorem:

If the polynomial P(x) is divided by D(x) = x − c, then

)()()()( xRxDxQxP

Plugging in x=c to the above equation one sees that

becomes

)())(()( xRcxxQxP

)())(()( cRcccQcP )()( cRcP

Example 6:

Let P(x) = x3 + 2x2 − 7.(a) Find the quotient and the remainder when P(x) is divided by x + 2.(b) Use the Remainder Theorem to find P(−2).

12 2 0

1

2

0

0

0

)(xQ )(xR

2)( xxQ

7

0

77)( xR

)()()()( xRxDxQxP 7272 223 xxxx

)2(P 7

Factor Theorem:

The number c is a zero of P(x) if and only if x−c is a factor of P(x); that is, P(x) = Q(x) · (x − c) for some polynomial Q(x). In other words, in Synthetic division the R(x) = 0 that is the last term is zero.

Example 7:

Use the Factor Theorem to determine whether x + 2 is a factor of f(x) = 3x6 + 2x3 − 176.

32 0 0

3

6

6

12

12

2

24

22

0 0

44

44

88

88

176176

0YES!!!!!!!!!!

Example 8:

Find a polynomial of degree 3 that has zeros 1, −2, and 3, and in which the coefficient of x2 is 3.

1x 2x 3xa

3222 xxxxa

322 xxxa 6233 223 xxxxxa

652 23 xxxa

aaxaxax 652 23

32 a

2

3

a

1x 2x 3x2

3