Multiplying Two Polynomials

When you are multiplying two polynomials, all you can ever multiply is a monomial times a monomial.

To multiply monomials, remember to do the following three steps:

  • Signs
  • Numbers
  • Variables, add exponents

We can multiply any two polynomials by repeatedly using the distributive property to multiply each term in the first polynomial by each term in the second polynomial.

Example:

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

Reasoning

 

2x(3x) + 2x(2) + 3(3x) + 3(2)
6x2 + 4x + 9x + 6
6x2 + 13x + 6

Use the distributive property to multiply 2x(3x) and 2x(2)
Repeat the distributive property to multiply 3(3x) and 3(2)

You may do this step mentally if you are comfortable with you mental math!

Multiply the monomials

2x(3x) = 6x2

2x(2) = 4x

3(3x) = 9x

3(2) = 6

Combine the like terms:

4x + 9x = 13x

Multiplying Two Binomials Using FOIL Method

To multiply two binomials, we will use a short cut of the distributive property called the FOIL Method.

FOIL Method for Multiplying Two Binomials

To multiply two binomials, find the sum of the product of:

F irst terms

O utside terms

I nside terms

L ast terms

Examples:

  1.   F    L   F     L
    (2x - 4)(3x + 1)
      O    I    I     O
    6x2 + 2x - 12x - 4
    6x2 - 10x - 4
    Reasoning
    Using FOIL, we multiply:
    F   2x(3x) = 6x2
    O  2x(1) = 2x
    I   -4(3x) = -12x
    L  -4(1) = -4

    Combine like terms:
    2x - 12x = -10x
    Usually, when multiplying binomials using FOIL, the outside terms and inside terms will be like terms and will be added together.
    To determine which terms we use for the F, O, I, and L, remember that a binomial has two terms; hence First term and Last term. Also, two binomials have a total of four terms. The Outside terms will be the first and fourth terms and the Inside terms will be the second and third terms.
    In each example, I will label the first (F) and last (L) terms above the binomial and the outside (O) and inside (I) terms below the binomial. You can do this also, if you need it as a reminder to yourself as to which terms to use.

  2.   F     L   F    L
    (5a + 2)(4a - 3)
      O     I     I   O
    5a(4a) + 5a(-3) + 2(4a) + 2(-3)
    20a2 - 15a + 8a - 6
    20a2 - 7a - 6
    Reasoning
    Using FOIL, we multiply:
    F   5a(4a) = 20a2
    O  5a(-3) = -15a
    I    2(4a) = 8a
    L   2(-3) = -6
    Combine like terms:
    -15a + 8a = -7a

  3.    F       L      F       L
    (4m2 - 5n2)(3m2 + 2n2)
       O       I        I       O
    4m2(3m2) + 4m2(2n2) - 5n2(3m2) - 5n2(2n2)
    12m4 + 8m2n2 - 15m2n2 - 10n4
    12m4 - 7m2n2 - 10n4
    Reasoning
    Using FOIL, we multiply:
    F   4m2(3m2) = 12m4
    O  4m2(2n2) = 8m2n2
    I   -5n2(3m2) = -15m2n2
    L  -5n2(2n2) = -10n4
    Combine like terms:
    8m2n2 - 15m2n2 = -7m2n2
     
    Notice when I multiplied -5n2(3m2), I wrote the answer as -15m2n2 (in alphabetical order). The commutative property tells us that -15n2m2 = -15m2n2.  We write terms with more than one variable in alphabetical order to help us identify like terms.

  4.   F     L   F     L
    (3x2 - 5)(2x + 3)
      O     I    I     O
    3x2(2x) + 3x2(3) - 5(2x) - 5(3)
    6x3 + 9x2 - 10x - 15
    Reasoning
    Using FOIL, we multiply:
    F  3x2(2x) = 6x3
    O  3x2(3) = 9x2
    I   -5(2x) = -10x
    L  -5(3) = -15

Because the first terms in the binomials are unlike terms, there are no like terms to combine.

Multiplying Polynomials Using Distributive Property

For all other multiplications involving polynomials, we must use the distributive property

We repeatedly use the distributive property to multiply each term in the first polynomial by each term in the second polynomial.

1.

(3x - 4)(2x2 - 7x + 5)

3x(2x2) + 3x(-7x) + 3x(5) - 4(2x2) - 4(-7x) - 4(5)

6x3 - 21x2 + 15x - 8x2 + 28x - 20

6x3 - 29x2 + 43x - 20

Reasoning

Use the distributive property to multiply by 3x:
3x(2x2) = 6x3
3x(-7x) = -21x
3x(5) = 15x

Use the distributive property to multiply by -4:
-4(2x2) = -8x2
-4(-7x) = 28x
-4(5) = -20

Combine like terms:
-21x2 - 8x2 = -29x2
15x + 28x = 43x

Unless otherwise stated, we will write our answers in descending order. If there is more than one variable, we will write our answer in alphabetical order as well as descending order.


Example: 5m3 - 3m2n + 2mn2 + 7n3

2. (4x2 - 7x + 5)(5x2 + 4x - 6)

Reasoning
  4x2(5x2) + 4x2(4x) + 4x2(-6) - 7x(5x2) - 7x(4x) - 7x(-6) + 5(5x2) + 5(4x) + 5(-6)
20x4 + 16x3 - 24x2 - 35x3 - 28x2 + 42x + 25x2 + 20x - 30
20x4 - 19x3 - 27x2 + 62x - 30 		Use the distributive property to multiply by 4x2:
4x2(5x2) = 20x4
4x2(4x) = 16x3
4x2(-6) = -24x2

Use the distributive property to multiply by -7x:
-7x(5x2) = -35x3
-7x(4x) = -28x
-7x(-6) = 42x

Use the distributive property to multiply by 5:
5(5x2) = 25x2
5(4x) = 20x
5(-6) = -30

Again, if you feel comfortable with your mental math, you do not have to write this step down.

Combine like terms:
16x3 - 35x3 = -19x3
-24x2 - 28x2 + 25x2 = -27x2
42x + 20x = 62x

Problem Solving:

Find the area of a rectangle (A = lw) if the length is five less than four times a number and the width is three more than twice the number.

A = lw

A =

A =

A =

A =

  F    L    F    L

(4n - 5)(2n + 3)

  O     I     I      O

4n(2n) + 4n(3) - 5(2n) - 5(3)

8n2 + 12n - 10n - 15

8n2 + 2n - 15

Reasoning

For the length, 5 less than tells us to subtract 5 from 4 times a number or 4n. So, 4n - 5

For the width, 3 more than tells us to add 3 to twice the number or 2n. Thus, 2n + 3

Using FOIL, we multiply lw:

F 4n(2n) = 8n2

O 4n(3) = 12n

I -5(2n) = -10n

L -5(3) = -15

Combine like terms:

12n - 10n = 2n

Try these problems

Solve

  1. (4a - 3)(3a + 4)
  2. (x + 5)(x + 7)
  3. (5p - 4)(3p + 2)
  4. (4m - 3n)(2m + 5n)
  5. (y2 - 2)(y + 3)
  6. (3x - 5)(2x2 + 7x + 3)
  7. (-2x + 7)(3x2 - 4x + 5)

    8. Problem Solving

  8. Find the area of a triangle (A = 1/2bh) if the base (b) of the triangle is seven less than five times a number and the height (h) is the total of three times the number and eight.

    9. Explain your reasoning.

Answers to Practice Problems

  1. (4a - 3)(3a + 4)
    12a2 + 16a - 9a - 12
    12a2 + 7a - 12

  2. (x + 5)(x + 7)
    x2 + 7x + 5x + 35
    x2 + 12x + 35

  3. (5p - 4)(3p + 2)
    15p2 + 10p - 12p - 8
    15p2 - 2p - 8

  4. (4m - 3n)(2m + 5n)
    8m2 + 20mn - 6mn - 15n2
    8m2 + 14mn - 15n2

  5. (y2 - 2)(y + 3)
    y3 + 3y2 - 2y - 6

  6. (3x - 5)(2x2 + 7x + 3)
    6x3 + 21x2 + 9x - 10x2 - 35x - 15
    6x3 + 11x2 - 26x - 15

  7. (-2x + 7)(3x2 - 4x + 5)
    -6x3 + 8x2 - 10x + 21x2 - 28x + 35
    -6x3 + 29x2 - 38x + 35

Reasoning

A = 1/2bh

A = 1/2(5n - 7)(3n + 8)

A = 1/2(15n2 + 40n - 21n - 56)

A = 1/2(15n2 + 19n - 56)

A = 15/2n2 + 19/2n - 28

The base is 7 less than which tells us to subtract 7 from five times a number 5n for 5n - 7 The height is a total which tells us to add 3 times a number 3n and 8 for 3n + 8

Multiply the two binomials using FOIL:

F  5n(3n) = 15n2

O  5n(8) = 40n

I  -7(3n) = -21n

L  -7(8) = -56

Combine like terms:

40n - 21n = 19n

Divide each term by 2.

15n2
  2

19n
 2

-56 = -28
  2