algebra 1 lesson 8-3 warm-up algebra 1 “multiplying binomials” (8-3) what is a binomial? how do...

ALGEBRA 1

Lesson 8-3 Warm-Up

ALGEBRA 1

“Multiplying Binomials” (8-3)

What is a binomial?

How do you multiply two binomials together?

Binomial: a polynomial that is the sum or difference of two terms that are not like terms

Examples: 2x + 1 9x4 + 11x -5x3 - 7

Method 1: Use an Area Model (write the product of each row and column in the box where they intersect, or cross)

Example: Simplify (2x + 3)(x + 4)

Create an area model in which each term has it own row or column.

Write the product of intersecting terms in the box where they intersect

2x2 + 8x + 3x + 12 Add the answers to each box

2x2 + 11x + 12 Combine like terms.

2x • x = 2x2 3 • x = 3x

2x • 4 = 8x 3 • 4 = 12

2x + 3

x

+

4

ALGEBRA 1


Method 2: Use the Distributive Property (horizontal method)


(2x + 3)(x + 4)

(2x • x) + (2x • 4) + 3x + 12

2x2 + 8x + 3x + 12

2x2 + 11x + 12 Combine like terms.

Tip: An easy way to remember what to multiply when using the Distributive Property to multiply binomials is to use FOIL, which is a mnemonic device (memory tool) like PEMDAS that stands for “First, Outer, Inner, Last”.

Example: Simplify (3x - 5)(2x + 7)

First + Outer + Inner + Last

(3x - 5)(2x + 7) + (3x - 5)(2x + 7) + (3x - 5)(2x + 7) + (3x - 5)(2x + 7)

6x2 + 21x + -10x + -35

6x2 + 11x - 35

Multiply the first term of the first binomail by each term of the second binomial. Then, multiply the second term of the first binomial by each term of the second binomial. Add these product together (don’t forget to combine like terms)

ALGEBRA 1


Method 3: Use the Distributive Property (vertical method): Multiply the two binomials together treating each term as a place value


2x + 3

x x + 4

8x + 12

2x2 + 3x

2x2 + 11x + 12

ALGEBRA 1


Method 4: Use an Area Model: Arrange the Algebra Tiles that model the binomial into the length and width of a rectangle. Then, find the area of the rectangle using Algebra Tiles (how many of each type it would take to completely fill the rectangle made by the length and width).


(2x + 3)(x + 4) = 2x2 + 11x + 12

x2 xx2 x x

xxxx

xxxx

1 1 11 1 11 1 11 1 1

x x 1 1 1

1111

x

x

x 11

Algebra Tiles Key

x 1

x2 x

x 1

ALGEBRA 1

Simplify (2y – 3)(y + 2).

(2y – 3)(y + 2) = (2y – 3)(y) Distribute 2y – 3.

= 2y2 – 3y Now distribute y and 2.

= 2y2 + y – 6 Simplify.

Multiplying BinomialsLESSON 8-3

Additional Examples

+ (2y – 3)(2)

+ 4y – 6

2y2y

-3

2

2y

-6

-3y

4y

1. Set up a table

2. Multiply terms in where rows and columns cross.

3. Combine like terms.

The product is 2y2 + y - 6.

ALGEBRA 1

Simplify (4x + 2)(3x – 6).

The product is 12x2 – 18x – 12.

Last

(2)(–6)+

Outer

(4x)(–6)+

Inner

(2)(3x)+

24x 6x 12– + –= 12x2

= 12x2 18x– 12–

First

= (4x)(3x)(4x + 2)(3x – 6)


Additional Examples

ALGEBRA 1

Find an expression for the area of the shaded region. Simplify.

area of outer rectangle (includes white area) = (3x + 2)(2x – 1)

area of inner rectangle (white area) = x(x + 3)

area of shaded region = area of outer rectangle – area of inner rectangle

= (3x + 2)(2x – 1) x(x + 3) Given areas.

= 6x2 Use FOIL to simplify (3x + 2) (2x – 1) and the Distributive Property to simplify

x(x + 3).

= 6x2 – 3x + 4x – 2 Group like terms.

= 5x2 – 2x – 2 Simplify.


Additional Examples

= 6x2 – 3x + 4x – 2 - (x2 + 3x) Shaded Area = Outer Area – Inner (White) Area

– 3x + 4x – 2 x2 + 3x

– x2 – 3x

ALGEBRA 1

Simplify the product (3x2 – 2x + 3)(2x + 7).

21x2 – 14x + 21 Multiply by 7.

6x3 – 4x2 + 6x Multiply by 2x.

Method 1: Multiply using the vertical method.

3x2 – 2x + 3

x 2x + 7

6x3 + 17x2 – 8x + 21 Add like terms.


Additional Examples

ALGEBRA 1

(continued)

Method 2: Multiply using the horizontal method.

= 6x3 – 4x2 + 6x + 21x2 – 14x + 21

The product is 6x3 + 17x2 – 8x + 21.

(2x + 7)(3x2 – 2x + 3)


Additional Examples

6x3

7

3

2x

3x2 -2x

-4x2

2121x2-14x

6x

= (2x)(3x2) – (2x)(2x) + (2x)(3) + (7)(3x2) – (7)(2x) + (7)(3)

= 6x3 + -4x2 + 21x2 + 6x + (-14x) + 21 Combine like terms.

ALGEBRA 1

Simplify each product using any method.1. (x + 3)(x – 6) 2. (2b – 4)(3b – 5)

3. (3x – 4)(3x2 + x + 2)

4. Find the area of the shaded region.

x2 – 3x – 18 6b2 – 22b + 20

9x3 – 9x2 + 2x – 8

2x2 + 3x – 1


Lesson Quiz

algebra 1 lesson 8-3 warm-up algebra 1 “multiplying binomials” (8-3) what is a binomial? how do...

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