Multiply Binomials


Multiply Binomials

You can use a visual model to multiply binomials. The model below shows that the product (x + 2)(3x + 1) equals


 x
 x
 x
 1
x
 x2
 x2
 x2
 x

x + 2
1
 x
 x
 x
 1
1
 x
 x
 x
 1

3x + 1

the solution for the binomial above is

3x2 + 7x +2

You can multiply two binomials using a table or a vertical method, 

Multiplying Binomials with a Table

Example

Find the product (-3x + 2)(8x + 7) and simplify.

Write the first polynomials on the left of the table and the second polynomial above the table


 8x
 7
-3x
 -24x2
 -21x
2
 16x
 14

Multiply to fill the table


Combine like terms

-24x2 - 21x + 16x + 14

-24x2 - 5x + 14

The answer is: -24x2 - 5x + 14

Multiplying Binomials Vertically

Example

Simplify the following expression: (2x - 5)(3x +  4)

Solution

Write and line up the first and second binomial

         2x  -  5

         3x  + 4

Multiply 4(2x - 5)

         2x  -  5

         3x  + 4

        8x  -  20

Multiply 3x(2x - 5) and line up like terms

         2x  -  5

         3x  + 4

        8x  -  20

6x2 - 15x       

Add 8x - 20 and 6x2 - 15x

         2x  -  5

         3x  + 4

        8x  -  20

6x2 - 15x       

6x2 - 7x - 20

The result is  6x2 - 7x - 20

Multiplying Binomials Horizontally

Example

You deposit $1 into a savings account with interest compounded annually. The balance of the account after two years can be found using the expression (1 + r)2, where r represents the interest rate. Expand this expression and simplify.

Solution

To expand the expression, multiply 2 binomials.

Expand (1 + r)2

(1 + r)2 = (1 + r)(1 + r)

then use the distributive property

(1 + r)(1 + r) = 1(1 + r) + r(1 + r) = 1 + r + r + r2

Simplify and write in standard form

1 + r + r + r2 = 1 + 2r + r2 = r2 + 2r + 1

Try it yourself

Find the product and simplify.

1.) (3t - 4)(t + 2)

2.) (x + 1)(x + 3)

3.) (b - 4)(b - 3)

The FOIL Method

The letters in the word FOIL can help you remember how to multiply binomials. The letters should remind you of the words

First, Outer, Inner, Last

F         L

(a + b) (c + d)

O

Multiplying with the FOIL Method

Find the product (2x + 3)(3x - 1) and simplify.

F
O
I
L
First
 +
 Outer
 +
 Inner
 +
 Last
2x 3x
 +
 2x (-1)
 +
 3 3x
 +
 3 (-1)
 Group terms
6x2
 +
 (-2x)
 +
 9x
 +
 (-3)
 Multiply
6x2 + 7x - 3
 Combine like terms

Try it yourself

Find the product and simplify.

4.) (d + 6)(d + 5)

5.) (x - 3)(x - 1)

6.) (5s + 3)(2s - 4)

7.) Complete the sentence. 
A polynomial with two terms is called a _?_.

8.) Describe and correct the error in the solution.

(x + 2)(x - 4) = (x x) + (x -4) + (2 x) + (2 (-4))

(x + 2)(x - 4) = (x x) + (x -4) + (2 x) + (2 (-4))

(x + 2)(x - 4) = x2 + 4x + 2x - 8

(x + 2)(x - 4) = x2 + 6x - 8

9.)

 

The middle of a baseball is a cork sphere with a radius of 0.6875 inches, and x is the thickness of the rubber and yarn coating surrounding the cork. 

Use the formula S = 4r2 and write a polynomial expression for the surface area of the baseball. Expand the expression and simplify.



Answer Key