# Multiplying Polynomials Algebra 1

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```					Name: ____________________________________                                                          Date: __________________
Multiplying Polynomials
Algebra 1

In the last lesson we worked extensively with multiplying polynomials by monomials. In this lesson
we will generalize this process so that we may multiply polynomials by polynomials. The first
exercise will illustrate the real number properties associated with this process.

Exercise #1: Fill in the blanks below with the real number property that justifies each step.

(1)   ( x + 2 )( x + 4 ) = x ( x + 4 ) + 2 ( x + 4 )
(2)                       = x⋅ x + x⋅4 + 2⋅ x + 2⋅4
(3)                       = x⋅ x + 4⋅ x + 2⋅ x + 2⋅4
(4)                       = x ⋅ x + ( 4 + 2) x + 2 ⋅ 4

= x2 + 6 x + 8

Exercise #2: Using real number properties, find the products given below.

(a) ( 2 x + 4 )( 3x − 1) =                     (b) ( x + 7 )( x − 5 ) =                 (c) ( 2 y − 3)( 4 y − 6 )

Multiplying two linear binomials is such an important skill that a mnemonic has been developed to
help remember it: FOIL – Multiply the First, Outer, Inner, and Last terms of the two binomials
together and then combine the like terms.

Exercise #3: Multiply the following binomials together either using a method as in Exercise #2 or by
“FOILing” the two binomials.

(a) ( x + 4 )( x + 1) =             (b) ( y + 3)( y − 5 ) =           (c) ( 2 x − 7 )( 3x + 2 ) =      (d) ( x − 5 )( x + 5) =

Exercise #4: Which of the following is equivalent to ( x − 4 ) ?
2

(1) x 2 + 16                  (3) x 2 − 8 x − 16

(2) x 2 − 16                  (4) x 2 − 8 x + 16

Algebra 1, Unit #6 – Quadratic Algebra – L5
The Arlington Algebra Project, LaGrangeville, NY 12540
We can also multiply polynomials together that have more than just two terms. Each term in the first
polynomial must multiply each term in the second polynomial for the distribution property to occur.

Exercise #5: Find the following product by distributing the binomial over the trinomial.

( 2 x − 3) ( 3 x 2 − 4 x + 9 ) =

Since multiplication of these higher powered polynomials can become confusing, it is helpful to use a
multiplication table to carry out the product.

Exercise #6: Use the following table to help evaluate the following product.

( x − 2 ) ( 3x 2 − 4 x + 7 ) =                                          3x2        −4x               7

x

−2

1.        ( x + 5 )( x − 2 ) =                                    7.      ( 3x + 1)2 =

2.        ( 2 x + 5 )( x + 3) =                                   8.      ( x + 6 )( x − 6 ) =

3.        ( x − 2 )( x + 3) =                                     9.      ( 2 x + 1)( 2 x − 1) =

4.        ( 3x − 5)( 2 x + 4 ) =                                  10.     ( 4 x + 5 )( 4 x − 5 ) =

5.        ( x − 5 )2 =
11.     ( 3x − 2 ) ( 2 x 2 + 5 x − 1)

6.        ( 2 x − 3)2 =

Algebra 1, Unit #6 – Quadratic Algebra – L5
The Arlington Algebra Project, LaGrangeville, NY 12540
Name: ____________________________________                                                Date: __________________
Multiplying Polynomials
Algebra 1 Homework
Skill
Find each of the following products in
simplest form.

1.    ( x − 2 )( x − 3) =                                 17.    ( x + 4 )( x − 4 ) =

2.    ( y + 6 )( y − 1) =                                 18.    ( x − 7 )( x + 7 ) =

3.    ( a + 5 )( a + 3) =                                 19.    ( y + 2 )( y − 2 ) =

4.    ( r + 4 )( r + 5) =                                 20.    ( 4 x + 3)( 4 x − 3) =

5.    ( 2 x − 3)( x + 5) =                                21.    ( 5 + x )( 5 − x ) =

6.    ( 2 x − 9 )( x + 3) =                               22.    ( 3 − y )( 3 + y ) =

7.    ( y − 7 )( y − 2 ) =                                23.    ( 4 x + 1)( 4 x − 1) =

8.    ( 2a + 5)( 3a + 7 ) =                               24.    ( x + 2 )2 =

9.    ( x − 3)( x + 8) =                                  25.    ( x − 6 )2 =
10.    ( x + 5)( x − 9 ) =                                26.    ( 4 x + 1)2 =
11.    ( 3x + 10 )( x − 5 ) =
27.    ( 3 x − 2 )2 =
12.    ( 5 + x )( x + 7 ) =
28.    ( 6 x + 1)2 =
13.    ( 6 − x )( 4 + x ) =
Applications
14.    ( 3 − 2 x )( 4 + 3x ) =                            29. If the side length of a square is given by the
binomial ( 4 x − 3) then which of the following
15.    ( 9 + x )( 8 + x ) =                                  gives the square’s area?

16.    ( 2 − x )( 3 − 5 x ) =                                   (1) 8 x + 6                 (3) 16 x 2 + 24 x + 9

(2) 16 x 2 − 9              (4) 16 x 2 − 24 x + 9

Algebra 1, Unit #6 – Quadratic Algebra – L5
The Arlington Algebra Project, LaGrangeville, NY 12540
Reasoning
30. Find the products of the following polynomials:

(
(a) ( 3 x + 5 ) 2 x 2 − 4 x + 3 = )                                              (            )
(b) ( x − 3) x 2 + 2 x + 9 =

31. Consider the following expression:                   ( x + 2 )3
(a) Rewrite the expression as a product of three binomials.

(b) Evaluate this product by multiplying the last two binomials in part (a) to form a trinomial and then
multiply this trinomial by the first binomial.

32. Rewrite the following expression without the use of parentheses. Keep in mind that you must
multiply the binomials together first and then perform the subtraction.

( x − 3)( x − 5 ) − ( x + 1)( x − 4 ) =

Algebra 1, Unit #6 – Quadratic Algebra – L5
The Arlington Algebra Project, LaGrangeville, NY 12540

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