Division and Factors

• When we divide one polynomial by another, we obtain a quotient and a remainder. If the remainder is 0, then the divisor is a factor of the dividend.

Example: Divide to determine whether

x + 3 and x 1 are factors of

3 22 5 4.x x x

Division and Factors continued

• Divide:

Since the remainder is –64, we know that x + 3 is not a factor.

2

3 2

3 2

2

2

5 20

3 2 5 4

3

5 5

5 15

2

rema

0 4

20 60

64 inder

x x

x x x x

x x

x x

x x

x

x

Division and Factors continued

• Divide:

Since the remainder is 0, we know that x 1 is a

factor.

2

3 2

3 2

2

2

4

1 2 5 4

5

4 4

4 4

0 remainder

x x

x x x x

x x

x x

x x

x

x

How do you divide a polynomial by another polynomial?

• Perform long division, as you do with numbers! Remember, division is repeated subtraction, so each time you have a new term, you must SUBTRACT it from the previous term.

• Work from left to right, starting with the highest degree term.

• Just as with numbers, there may be a remainder left. The divisor may not go into the dividend evenly.

The Remainder Theorem

If a number c is substituted for x in a polynomial f(x), then the result f(c) is the remainder that would be obtained by dividing f(x) by x c. That is, if f(x) = (x c) • Q(x) + R, then f(c) = R.

Synthetic division is a “collapsed” version of long division; only the coefficients of the terms are written.

Synthetic division is a quick form of long division for

polynomials where the divisor has form x - c. In synthetic

division the variables are not written, only the essential part

of the long division.

652 2 xxxx

xx 22 _______x3 6

3

63 x_______

6 5 1 2

1

-2

3

-60

22 xx

0

quotient

remainder

2 5 6 ( 2)( 3)x x x x

ExampleUse synthetic division to find the quotient and remainder.

The quotient is – 4x4 – 7x3 – 8x2 – 14x – 28 and the remainder is –6.

5 4 3 24 6 2 50 ( 2)x x x x x

–6–28–14–8–7–4

–56–28–16–14–8

500261–42 Note: We must write a 0 for the missing term.

Example continued:written in the form

5 4 3 24 6 2 50x x x x

( ) ( ) ( ) ( )P x d x Q x R x

4 3 22 -4x 7 8 14 28 6x x x x

By the remainder theorem

we know (2) 6P

Example• Determine whether 4 is a zero of f(x), where

f(x) = x3 6x2 + 11x 6.

• We use synthetic division and the remainder theorem to find f(4).

• Since f(4) 0, the number is not a zero of f(x).

63–21

12–84

–611–614

The Factor Theorem

• For a polynomial f(x), if f(c) = 0, then x c is a factor of f(x).

Example: Let f(x) = x3 7x + 6. Solve the equation f(x) = 0 given that x = 1 is a zero.

Solution: Since x = 1 is a zero, divide synthetically by 1.

Since f(1) = 0, we know that x 1 is one factor and the quotient x2 + x 6 is another.So, f(x) = (x 1)(x + 3)(x 2).For f(x) = 0, x = 3, 1, 2.

0-611

-611

6-701 1

Factor Theorem

• f(x) is a polynomial, therefore f(c) = 0 if and only if x – c is a factor of f(x).

• If we know a factor, we know a zero!

• If we know a zero, we know a factor!

• Definition of Depressed Polynomial

• A Depressed Polynomial is the quotient that we get when a polynomial is divided by one of its binomial factors

• Which of the following can be divided by the binomial factor (x - 1) to give a depressed polynomial (x - 1)?Choices: A. x2 - 2x + 1B. x2 - 2x - 2C. x2 - 3x - 3D. x2 - 2

Using the remainder theorem to find missing coeffecients…

• Find the value of k that results in a remainder of “0” given…

)3()243( 23 xkxxx