Algebra 1 > Notes > Yorkcounty Final > Yc > Lesson 13

Multiplying Binomials

Lesson 13

Again we will be using the tiles as described below to complete binomial multiplication. Remember a binomial has two monomials Area = x2

x x

x 1 1

1

Area = x

Area = 1

Example: Manipulative Activity Use algebraic tiles to model and simplify (2x +3) (x + 3) First, use the tiles to model the two binomials 2x +3, and x +3

x+3

2x+3

Now we will fill in the large rectangle with the tiles in the same way that we would a multiplication table. The first spot to be filled is multiplying a “1” square times a “1” square, 1 x 1 = 1. They are highlighted in yellow. Thus we must fit in a 1 square….

x+3

x+3

2x+3

2x+3

If we continue this process we will multiply more 1 squares times 1 squares as is shown below.

x+3 Until all the green squares are filled in.

2x+3 x+3

2x+3

Now we must multiply the x squares times the 1 squares.

x+3 Until we get this result...

2x+3 x+3

2x+3

Finally we must multiply x times x to get x2

x+3

Then we get this rectangular result.

2x+3 x+3

2x+3

x+3

If we analyze our result we see that we have…..

2x+3 2

+

9

+

9

or 2

( x2 )

+

9 (x) or

2x2 + 9x + 9

Notice that when we multiply two binomials: (2x +3)(x+3)

+

9 (1)

we get a trinomial answer: 2x2 + 9x + 9

Now you try it: 1) Using tile pictures, express the product of (3x + 1) (x + 4) as a trinomial.

Area = x2

x x

x 1

1

Area = x

Area = 1

1 Advance to the next slide to check your answer when you are done….

Try it solution: (3x + 1) (x + 4)

x+4

3x + 1

x+4

3x + 1

Answer: 3x2 + 13x + 4

Now that you can use tiles to multiply binomials, let’s learn how to use algebraic techniques of distribution to multiply them. Let’s use the example of (2x +4)(x+3) You are used to doing problems like 2x (x+3) using distribution. Simply multiply both terms in the parentheses by 2x 2x (x) + 2x (3) = 2x2 + 6x Now you have two things to distribute through the (x + 3), there is a 2x in the first set of ( ) and a 4. You can think of multiplying a binomial times a binomial as a “double-distribution” problem. Let’s complete the example: (2x + 4) (x + 3) 2x (x + 3) +4 (x +3) 2x2 + 6x + 4x + 12 2x2 +

10x

+ 12

Combining like terms in the middle

EXAMPLE 1: Express the product of (2x - 1) (3x - 4) as a trinomial.

A)6 x + 4 2

B )6 x + 11x + 4 2

C )6 x − 11x + 4 2

D) 6 x − 4 2

Advance to the next slide to see the answer worked out….

(2x - 1) (3x - 4) 2x (3x -4) +2x·3x 6x2 6x2

-1(3x -4)

+2x·-4

Don’t forget the sign in front of a number goes with it. See -1

-1·3x

-1·-4

-3x

+4

-8x -11x

+4

Combining Like Terms of -8x and -3x

The best answer choice is C) 6x2 -11x +4

Try This: 2) Express the product of (3x -4) (2x + 5) as a trinomial using algebra.

Advance to the next slide to check your answer when you are ready.

Try This Solution: (3x - 4) (2x + 5) 3x (2x + 5) +3x•+2x +3x•+5 6x2 6x2

+15x

-4 ( 2x + 5) -4•+2x -8x

+7x

6x2 +7x -20

-4•+5 -20 -20

Applications of Multiplying Binomials: One example is finding the area of a square or rectangle. Remember that area is equal to length times width. A = LW

So, if the length and width are given as binomials you will have to perform double distribution in order to find the area…

See the next example

Example 2: The length of the side of a square is 4x -3. Find the area of the square in terms of x.

F )16 x + 24 x + 9 2

G )16 x − 12 x + 9 2

H )16 x + 9 2

J )16 x − 24 x + 9 2

Recall that a square has equal length and width, so we can label our picture as below.

A= 4x - 3

L • (4x - 3)

•

+4x(4x -3) 4x - 3 Now we will use the area formula along with our new skill of multiplying binomials to get an expression for the area in terms of x.

W (4x - 3) -3(4x -3)

+4x(4x) +4x(-3) -3(4x) -3(-3) +16x2 -12x 16x2

-12x

-24x

+9 +9

Hence, the best answer choice is J) 16x2 -24x +9.

Try This: 3) A squares side length is 5x - 2. Express the area of the square in terms of x. 5x-2

4) Find the area of a rectangle with a length of 2x -7 and a width of 3x + 4. 2x-7 Advance to next slide when you are ready to check your answers.

3x+4

Try This Solutions: 3) A =

(L)

(W)

(5x-2)

(5x-2)

5x (5x-2) -2(5x -2)

4) A =

(L)

(W)

(2x -7)

(3x +4)

2x(3x+4) -7(3x +4) 2x(3x)+2x(+4) -7(3x) -7(+4)

5x(5x)+5x(-2) -2(5x)-2(-2)

6x2

25x2

6x2

25x2

- 10x

- 10x + 4

-20x

+8x

-21x

-13x

+4

6x2 -13x -28 25x2 -20x +4

- 28 -28

Some tile problems may ask you to work backwards and find the two binomials that were multiplied together to get a particular trinomial. This is the goal of the next example.

Example 3: A rectangle with an area of 2x2+5x+3 is modeled below using algebraic tiles. What are the dimensions of the rectangle in terms of x?

A)( x + 4)by ( x − 1) B)(2 x + 1)by ( x − 1) C )(2 x + 1)by ( x + 3) D)(2 x + 3)by ( x + 1)