Warm-Up Multiplying Monomials and Binomials

Home , Commutative property

? Lesson Question

Lesson Goals

Multiply and binomials.

Apply the Identify special products of

Determine products . property. using

models.

W 2K Words to Know

Fill in this table as you work through the lesson. You may also use the glossary to help you.

the result of a multiplication of two or more terms

a number that evenly divides into another number; a polynomial that evenly divides into another polynomial an expression involving a sum of powers in one or more variables multiplied by coefficients, where the powers must be whole numbers the property stating that the product of a factor times a given quantity containing a sum or difference is equal to the sum or difference of the products of that factor times each term from within the quantity

Using the Distributive Property

Use the distributive property to find Distributive property: the product.

−4 푥 − 5 푎 푏 + 푐 = −4 푥 + −5 푥 + −4 −5 −4푥 + 20

Slide 2 Multiplication Properties of Polynomials

The following multiplication properties are true for any polynomials a, b, and c:

Closure property The of two polynomials is a polynomial.

Commutative property 푎푏 =

Associative property (푎푏)푐 = 푎(푏푐)

property 푎(푏 + 푐) = 푎푏 + 푎푐

Using the Distributive Property

Multiply: 6푦3(−5푦 + 3) • Product of powers rule:

3 3 6푦 −5푦 + (6푦 )(3) 푎푚 ∙ 푎푛 = + 18푦3

4 Multiplying a Monomial by a Binomial Using Algebra Tiles

2푥 푥 − 2 = −

Slide 8 Multiplying Two Binomials Using Algebra Tiles

Draw the missing algebra tiles to complete the model.

2푥 − 1)(푥 − 4 2푥2 − +

푥 − 3 푥 + 2 − 3푥 + 2푥 −

푥2 − 푥 − 6

Slide 10 How to Multiply a Binomial by a Binomial Using a Model

How to multiply polynomials using a model: 1. Write each factor as headers to a row or column.

2. Multiply each entry by and column.

3. Add the terms inside the model, combining when

necessary.

4. Write the resulting polynomial in form.

Multiplying a Binomial by a Binomial Using a Table

1. Write each factor as headers to a row Multiply: −2푣 + 1)(3푣 − 5 or column. 2. Multiply each entry by row and 3푣 column. 3. Add the terms inside the model, combining like terms when −6푣2 necessary. 4. Write the resulting polynomial in standard form. +1 −5

= −6푣2 + −5

Slide 12 Multiplying a Binomial by a Binomial

Use the property to find the product of −7푛 − 4 and 3 + 2푛.

−7푛 − 4 3 + 2푛

−7푛 + −7푛 + −4 + −4 2푛

−21푛 + −14푛2 + −12 + −8푛

−14푛2 − − 12

14 Multiplying Multivariable Polynomials Find the product of the binomials: (2푦 − 푥)(푦 − 푥) 2푦 + 2푦 + −푥 + −푥

2푦2 + −2푥푦 + + (푥2) 푥2 − 3푥푦 +

Slide 16 Difference of Squares

• Difference of squares: Multiply: −2푥 + 8 −2푥 − 8 푎 = −2푥 푎 + 푏 푎 − 푏 = 푏 = 8 푎2 − 푎푏 + 푎푏 − 푏2 ( )2 − ( )2 푎2 − 푏2 − 64

18 Perfect Square Trinomials ADDITION • Perfect square trinomial:

(푎 + 푏)2 = (푎 + 푏)(푎 + 푏) =

Example: Multiply: (7푥 + 2)(7푥 + 2)

푎 = 푏 = 2 (7푥)2 + 2 7푥 2 + (2)2 + 28푥 + 4

Perfect Square Trinomials SUBTRACTION • Perfect square trinomial:

(푎 − 푏)2 = (푎 − 푏)(푎 − 푏) =

Example: Multiply: 푔 − 10 2

푎 = 푏 = 10 ( )2 − 2 10 + (10)2 − 20푔 + 100

? Lesson Question What does the product of polynomials look like?

Answer

Slide 2 Review: Key Concepts

• Multiplying a by a binomial or binomial by a binomial can be

done by using the distributive property, a table, or algebra tiles.

• Special products:

• Difference of squares: 푎 + 푏 푎 − 푏 = 푎2 − 푏2

• Perfect trinomials

(푎 − 푏)2= 푎2 − 2푎푏 + 푏2 (푎 + 푏)2= 푎2 + 2푎푏 + 푏2

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