Multiplying and Dividing Polynomials

Tammy WallaceVarina High

RecallRecall a monomial is a term with just term, a number, or the product of numbers and variables.

A BINOMIAL is the sum or difference of two monomials.

A TRINOMIAL is the sum or difference of three monomials.

Multiplying Polynomials

MULTIPLYING POLYNOMIAL can be represented several different ways.

Monomials x Binomial Monomial x Trinomial   

 

Binomial x Binomial Binomial x Trinomial   

 

𝟐(𝒙+𝟒) x

(𝒙+𝟑)(𝒙−𝟐) )

Multiplying Polynomials

When multiplying polynomials use the distributive property to completely simplify each expression.

MONOMIALS X BINOMIALS

Use the box method

  x 4

2  

 𝟐 𝒙𝟖 2 (𝑥+4 )=¿𝟐 𝒙+𝟖

Use the box method

  x -1

3  

 3 −𝟑 3 (𝑥−1 )=¿ 3

Monomial x TrinomialSimplify

Use the box method

  x -y

2x

 

 𝟐 𝒙𝟐−𝟐 𝒙𝒚

2 𝑥 (𝑥− 𝑦+5 )=¿𝟐 𝒙𝟐−𝟐 𝒙𝒚 +𝟏𝟎𝒙

𝟏𝟎𝒙

Monomial x TrinomialSimplify

Use the box method

  -2x  

 

−𝟐 𝒙𝟐𝒚𝟑𝟐 𝒙𝟐 𝒚𝟐

−𝟐 𝒙𝟑𝒚𝟐+𝟐 𝒙𝟐𝒚𝟐−𝟒 𝒙 𝒚𝟐

−𝟒 𝒙𝒚𝟐

Binomial x BinomialSimplify

Group all like terms and combine for the final answer.

Use the box method

  x 3

x  

 

7

𝒙𝟐 3 𝒙7 21

Binomial x BinomialSimplify

Group all like terms and combine for the final answer.

  3x -5

5x  

 

2

𝟏𝟓𝒙𝟐 −𝟐𝟓𝒙

6 -10

Binomial x BinomialSimplify

Group all like terms and combine for the final answer.

  2x -1

2x  

 

1

𝟒 𝒙𝟐 −𝟐 𝒙

2 -1

Special BinomialsSimplify This binomial is being squared. When anything is raised to a power,

= =

  x -3

x  

 

-3

𝒙𝟐 −𝟑 𝒙

-3 9

multiply the binomial by itself based on the outside exponent.

Trinomial x TrinomialSimplify

  -3x  

 

𝟐 𝒙𝟑−𝟔 𝒙𝟐 𝟖 𝒙

¿2 𝑥3−6𝑥2+2 𝑥2+8𝑥−3 𝑥+4

𝟐 𝒙𝟐 −𝟑 𝒙 𝟒

¿𝟐 𝒙𝟑−𝟒 𝒙𝟐+𝟓 𝒙+𝟒

Trinomial x TrinomialSimplify

  -5x  

 

𝟐 𝒙𝟑−𝟏𝟎𝒙𝟐 𝟖 𝒙

¿2 𝑥3−10 𝑥2−5𝑥2+8 𝑥+25 𝑥−20

-5 25 −𝟐𝟎

¿𝟐 𝒙𝟑−𝟏𝟓𝒙𝟐+𝟐𝟑𝒙−𝟐𝟎

Dividing PolynomialsWhen dividing polynomials, rewrite the expression by breaking it up based on the number of terms in the numerator.

After breaking up the expression, simplify each term.

Dividing PolynomialsProblem Break up Simplify

 

    

Problem Break up Simplify 

  𝟓 𝒚𝟐+𝟐 𝒚−𝟏

Dividing PolynomialsSometime division can be expressed like . Rewrite the problem as a fraction and solve like normal.

Problem Rewrite Break up Simplify

   

   2 2x 2x 32

2 2x 2x 32

2𝑥2+322 𝑥2

2𝑥2

2𝑥2+ 322 𝑥2 𝟏+

𝟏𝟔𝒙𝟐