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Dividing Polynomials & The Remainder Theorem

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Page 1: Dividing Polynomials & The Remainder Theorem. Dividing Polynomials When dividing a polynomial by a monomial, divide each term in the polynomial by the

Dividing Polynomials &

The Remainder Theorem

Page 2: Dividing Polynomials & The Remainder Theorem. Dividing Polynomials When dividing a polynomial by a monomial, divide each term in the polynomial by the

Dividing Polynomials

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

• Ex.

xy

yxxyyx

4

1284 32223

Page 3: Dividing Polynomials & The Remainder Theorem. Dividing Polynomials When dividing a polynomial by a monomial, divide each term in the polynomial by the

Dividing Polynomials

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

• Ex.

122 5934242 tttxxx

Page 4: Dividing Polynomials & The Remainder Theorem. Dividing Polynomials When dividing a polynomial by a monomial, divide each term in the polynomial by the

More Dividing

2

810135235

2312

x

xxxaaa

Page 5: Dividing Polynomials & The Remainder Theorem. Dividing Polynomials When dividing a polynomial by a monomial, divide each term in the polynomial by the

Dividing

113124254 524 mmmyyyy

Page 6: Dividing Polynomials & The Remainder Theorem. Dividing Polynomials When dividing a polynomial by a monomial, divide each term in the polynomial by the

Dividing, one more

2628156 34 xxxx

Page 7: Dividing Polynomials & The Remainder Theorem. Dividing Polynomials When dividing a polynomial by a monomial, divide each term in the polynomial by the

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

--where a is a constant ax

_____________________

8101352

2810135 23

xxxx

Page 8: Dividing Polynomials & The Remainder Theorem. Dividing Polynomials When dividing a polynomial by a monomial, divide each term in the polynomial by the

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

Page 9: Dividing Polynomials & The Remainder Theorem. Dividing Polynomials When dividing a polynomial by a monomial, divide each term in the polynomial by the

More synthetic Division

32314174 24 xxxx

Page 10: Dividing Polynomials & The Remainder Theorem. Dividing Polynomials When dividing a polynomial by a monomial, divide each term in the polynomial by the

The Remainder Theorem

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

Page 11: Dividing Polynomials & The Remainder Theorem. Dividing Polynomials When dividing a polynomial by a monomial, divide each term in the polynomial by the

Example

Given , find

Page 12: Dividing Polynomials & The Remainder Theorem. Dividing Polynomials When dividing a polynomial by a monomial, divide each term in the polynomial by the

Example

Given , find

Page 13: Dividing Polynomials & The Remainder Theorem. Dividing Polynomials When dividing a polynomial by a monomial, divide each term in the polynomial by the

Example

Given , find

Page 14: Dividing Polynomials & The Remainder Theorem. Dividing Polynomials When dividing a polynomial by a monomial, divide each term in the polynomial by the

The Factor Theorem

Let be a polynomial.

a. If then is a factor of

b. If is a factor of then

Page 15: Dividing Polynomials & The Remainder Theorem. Dividing Polynomials When dividing a polynomial by a monomial, divide each term in the polynomial by the

Is a factor of

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

Page 16: Dividing Polynomials & The Remainder Theorem. Dividing Polynomials When dividing a polynomial by a monomial, divide each term in the polynomial by the

Use the Factor Theorem to Solve

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

Page 17: Dividing Polynomials & The Remainder Theorem. Dividing Polynomials When dividing a polynomial by a monomial, divide each term in the polynomial by the

Solve

given that -1 is a zero of the function.


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