Section 2.4 Dividing Polynomials; Remainder and Factor Theorems · 2019-12-30 · Use synthetic division and the remainder theorem to find the indicated function value. f x x x( )

Section 2.4

Dividing Polynomials;

Remainder and Factor Theorems

Long Division of Polynomials

and

The Division Algorithm

Long Division of Polynomials

Long Division of Polynomials

2

3 2 9 6 5x x x

12 5x

4

13

3x

29 6x x

12 8x

13

3 2x

Long Division of Polynomials with Missing Terms

2 3

3 2

2

2

x +5x -3 x 3x 2

x +5x 3x

-5x 6x 2

-5x 25x 15

31x- 17

5x 2

31 17

5 3

x

x x

You need to leave a hole when you have

missing terms. This technique will help

you line up like terms. See the dividend

above.

Example

Divide using Long Division.

73652 23 xxx

Example

Divide using Long Division.

2 4 32 1 8 3 +5 1x x x x x

Dividing Polynomials Using

Synthetic Division

Comparison of Long Division and Synthetic

Division of X3 +4x2-5x+5 divided by x-3

Steps of Synthetic Division dividing 5x3+6x+8 by x+2

Put in a 0 for the missing term.

2 5 + 7 - 1

Using synthetic division instead of long division.

Notice that the divisor has to be a binomial of

degree 1 with no coefficients.

5

10

3

6

5

2

55 3

2

2 5 7 1

xx

x x x

Thus:

Example

Divide using synthetic division.

3 23 5 7 8

4

x x x

x

The Remainder Theorem

If you are given the function f(x)=x3- 4x2+5x+3 and

you want to find f(2), then the remainder of this

function when divided by x-2 will give you f(2)

f(2)=5

2(1) for f(x)=6x 2 5 is

1 6 -2 5

6 4

6 4 9

f(1)=9

f x

Example

Use synthetic division and the remainder

theorem to find the indicated function value. 3 2( ) 3 5 1; f(2)f x x x

The Factor Theorem

Solve the equation 2x3-3x2-11x+6=0 given that 3 is

a zero of f(x)=2x3-3x2-11x+6. The factor theorem

tells us that x-3 is a factor of f(x). So we will use

both synthetic division and long division to show

this and to find another factor.

Another factor

Example

Solve the equation 5x2 + 9x – 2=0 given

that -2 is a zero of f(x)= 5x2 + 9x - 2

Example

Solve the equation x3- 5x2 + 9x - 45 = 0 given

that 5 is a zero of f(x)= x3- 5x2 + 9x – 45.

Consider all complex number solutions.

(a)

(b)

(c)

(d)

3 2Divide 2 8 3x x x x

2

2

2

2

8

4 2

4 14

344 14

3

x x

x x

x x

x xx

(a)

(b)

(c)

(d)

3 2

Use Synthetic Division and the Remainder

Theorem to find the value of f(2) for the function

f(x)=x +x - 11x+10

2

0

5

12