Multiplying PolynomialsLesson 2-2

Warm-upPerform the polynomial operation.1. (x2 + 5x – 3) + (x3 – 2x2 + 7)2. (5x – 3 + 2x2) + (4 – 5x2 + x)3. (x2 + 5x – 3) – (x3 – 2x2 + 7)4. (5x – 3 + 2x2) – (4 – 5x2 + x)

Challenge:(4x – 2) – 2(3 – 7x2 – x)

RefresherSimplify each of the following.

1. 3(x + 5)

2. 7(m – 3)

3. x2 • x3

4. 4x2 • x3

5. 4x2 • 5x3

= 3x + 15

= 7m – 21

= x5

= 4x5

= 20x5

RefresherPerform the polynomial operation.1. (5x – 3)(2x + 7)2. (x – 3)(4 + x)3. (11x + 2)(x + 7)4. (-x – 5)(3x + 4)

Challenge:(4x – 2)(7x2 – x - 3)

Polynomial MultiplicationWhen we are multiplying polynomials, we are

applying the distributive property of multiplication over addition. Pay attention that we only distribute when there is addition happening within the parentheses for which we are multiplying.

Example: 2(5x + 3) – use distributive property

10x + 6 5(5xyz) – do not use distributive

property25xyz

Multiplication of Monomial and PolynomialTo multiply a polynomial by a monomial, we

distribute the monomial to all terms in the polynomial.

Example: 6x(x2 + 5x – 3)6x3 + 30x2 – 18x

Let’s PracticeMultiply the following polynomials using

distribution.1. 2x(x + 4) 2. 3x2(-2x + x2 – 7)3. 2xy(x2 – 5xy + 3y2)

Challenge:3x(x + 4) + 5(x + 4)

MultiplicationDo the following multiplication operations.1. 62 3. 112

x 35_ x 17_

2. 372 4. 535x 41_ x 81

MultiplicationWhen we are multiplying multi-digit numbers

together, we can consider the process similar to that of distribution. We must make sure that we multiply each part of each number by every part of the other number.

Example: (35)(62) = (30 + 5)(60 + 2)We have the multiply the 30 by both the 60 and the

2, then multiply the 5 by the 60 and the 2. Finally, we add all of our components to get a final answer.

30(60 + 2) + 5(60 + 2) = 30(60) + 30(2) + 5(60) + 5(2)1800 + 60 + 300 + 10 = 2170

Multiplying BinomialsWhen we multiply binomials, we need to multiply

each part of the binomial by each of the other binomial. (You may hear the acronym FOIL to describe this process, but this only works for binomials)

Example: (2x + 5)(x + 2)First: (2x)(x) = 2x2

Outer: (2x)(2) = 4xInner: (5)(x) = 5xLast: (5)(2) = 10

Finally, we add all the terms: 2x2 + 4x + 5x + 10 = 2x2 + 9x + 10

Multiplying with a Grid(3x – 5)(2x + 2)

Multiply PolynomialsWith polynomials, we need to make sure that

we multiply each term by each other. As such, FOILing no longer works because we have more than 2 terms per polynomial.

If we set up the problem like a multi-digit multiplication problem, we can progress the same way we always have.

Multiplying Polynomials 2(9 6)(5 2)x x x

29 6x x 5 2x

122x218x030x25x

345x 1232x223x

345x

Find the Product

Multiplying Polynomials with a GridFind the product (9x2 – x + 6)(5x – 2)

Let’s PracticeMultiply the following polynomials.1. (m + 3)(5m – 4)

2. (2k – 3)(k2 + 7k – 8)

3. (a2 + 5a – 4)(2a + 3)

Challenge:(x2 + 3x + 5)(x2 – 3x + 5)

Area Model

Example of Area ModelFind the product of (x + 4) and (x + 8) using

an area model.

Multiplying Polynomials using the area of the polynomial shown

You know that the area of a rectangle is the product of its length and width. In the model, let 3x + 1 represent the length and let x + 2 represent the width. To find the total area of the model, add the areas of

each rectangular part. 2 2 2(3 1)( 2) 1 1A l w x x x x x x x x x x x x

--------------- 3x + 1 ----------

----

--

----

X + 2

xx x

x

xxx

x2x2x2

11

1

1

xxx

x

1

23 7 2x x

Multiplying Polynomials using a Volume Model

Write a polynomial for the volume of the rectangular prism shown.

You know that the volume of a rectangular prism is the product of its length, width, and height. In the figure shown, let x represent the length, x + 1 represent the width and x + 2 represent the height.

volume length width height

x + 2

xx + 1

( 1)( 2)x x x 1x 2x 22x0x2x23x2x

2( 3 2)x x x 2( )x x (3 )x x (2)x

3 23 2x x x

The square of a binomial pattern2 2 2

2 2 2

( ) 2

( ) 2

a b a ab b

a b a ab b

The square of a binomial patterna. (7x + 2)2 b. (3x – 2)2

2(7 2)x 2(7 )x 2(7 )(2)x 2(2)249x 28x 4

2(3 2)x 2(3 )x 2(3 )(2)x 2(2)29x 12x 4

Let’s Practice2

2

2

2

( 9)

(3 7)

(2 3)

(10 4)

y

x

w

r

Sum and Difference Patterns

2 2( )( )a b a b a b

Sum and Difference Patterns

a. (m + 9)(m – 9)b. (4n – 3)(4n + 3)2 2( 9)( 9) 9m m m 2 81m

2 2(4 3)(4 3) (4 ) 3n n n 216 9n

Let’s Practice

( 11)( 11)

(7 1)(7 1)

(2 9)(2 9)

(6 8)(6 8)

g g

f f

h h

k k

Expand a binomialExpand the following: (x + 2)3.

Step 1: Rewrite the problem without an exponent.(x + 2)(x + 2)(x + 2)

Step 2: Multiply the first two binomials to get a trinomial.

(x2 + 4x + 4)(x + 2) (x+2)(x2+4x+4)

Step 3: Multiply your new trinomial by the other binomial.

x3 + 4x2 + 4x + 2x2 + 8x + 8Step 4: Combine like terms.

x3 + 6x2 + 12x + 8

Let’s PracticeExpand the following binomial:

(x – 3)3

HomeworkPg. 88-89, #18-27, 30, 48