Dividing Polynomials &

The Remainder Theorem

Dividing Polynomials

• When dividing a polynomial by a monomial, divide each term in the polynomial by the monomial

• Ex.

xy

yxxyyx

4

1284 32223

Dividing Polynomials

• When dividing a polynomial by a polynomial, use long division (called division algorithm)

• Ex.

122 5934242 tttxxx

More Dividing

2

810135235

2312

x

xxxaaa

Dividing

113124254 524 mmmyyyy

Dividing, one more

2628156 34 xxxx

Synthetic Division—shortcut to long division—only use if dividing by

--where a is a constant ax

_____________________

8101352

2810135 23

xxxx

More synthetic Division

• Remember if you are missing a term, you need to put in a zero for the coefficient

12

25

593

2883

ttt

zzz

More synthetic Division

32314174 24 xxxx

The Remainder Theorem

If you divide a polynomial of degree by then the remainder is

Example

Given , find

Example

Given , find

Example

Given , find

The Factor Theorem

Let be a polynomial.

a. If then is a factor of

b. If is a factor of then

Is a factor of

If it is, write as a product of two factors.

Use the Factor Theorem to Solve

Solve the equation given that 3 is a zero of the function.

Solve

given that -1 is a zero of the function.