2.4_dividing polynomials.pdf

8/11/2019 2.4_Dividing Polynomials.pdf

1/25

Section 2.4

Dividing Polynomials;Remainder and Factor Theorems


2/25

Long Division of Polynomials

and

The Division Algorithm


3/25


4/25



5/25


2 3 2 9 6 5x x x

12 5x

4

13

3x

29 6x x

12 8x

13

3 2x


6/25

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 231 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.


7/25


8/25

Example

Divide using Long Division.3 22 5 6 4 +7x x x


9/25

Example

Divide using Long Division.2 4 32 1 8 3 +5 1x x x x x


10/25

Dividing Polynomials Using

Synthetic Division


11/25


12/25

Comparison of Long Division and Synthetic

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


13/25

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

Put in a 0 for the missing term.


14/25

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

22 5 7 1

x

xx x x

Thus:


15/25

Example

Divide using synthetic division.3 23 5 7 8

4

x x x

x


16/25

The Remainder Theorem


17/25

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


18/25

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

1 6 -2 5

6 4

6 4 9

f(1)=9

f x


19/25

Example

Use synthetic division and the remaindertheorem to find the indicated function value.3 2( ) 3 5 1; f(2)f x x x


20/25

The Factor Theorem


21/25

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


22/25

Example

Solve the equation 5x2

+ 9x

2=0 giventhat -2 is a zero of f(x)= 5x2 + 9x - 2


23/25

Example

Solve the equation x3

- 5x2

+ 9x - 45 = 0 giventhat 5 is a zero of f(x)= x3- 5x2+ 9x 45.

Consider all complex number solutions.


24/25

(a)

(b)

(c)

(d)

3 2Divide 2 8 3x x x x 2

2

2

2

8

4 24 14

344 14

3

x x

x xx x

x xx


25/25

(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

2.4_dividing polynomials.pdf

Documents