Multiplying Binomials Using the Punnett Square

In this blog post, I'll show you how to multiply two binomials with any amount of terms together using the Punnett Square Method, or should I say, Punnett Rectangle Method.

This is best illustrated by starting with two binomials, for instance consider the following example:

Example 1

Multiply the following binomials: (3x - 1)(x + 5).

Before we talk about the Punnett Square method of multiplying these binomials let's look at two of the more traditional methods: FOIL and double distribution.

FOIL

FOIL stands for first, outer, inner, last. In other words, you multiply the first terms together, then the outer terms, then the inner terms before finally multiplying the last terms together.

Let's multiply (3x - 1)(x + 5) using FOIL.

  • F - 3x · x = 3x2
  • O - 3x · 5 = 15x
  • I - −1 · x = −x
  • L - −1 · 5 = −5

We then combine like terms to get 3x2 + 14x - 5.

Double Distribution

Double Distribution has an advantage over FOIL because you can use it not only for binomial multiplication, but for any type of multiplication.

To complete double distribution break apart the front parenthesis and multiply each term times the second parenthesis:

3x(x + 5) + −1(x + 5) = 3x · x + 3x · 5 + −1 · x + −1 · 5 = 3x2 + 15x − x − 5 = 3x2 + 14x - 5.

Using the Punnett Square Method

To use the Punnett Square Method first draw a Punnett Square:

       
       

Then set each side of the Punnett Square each to a binomial as follows:

3x −1
x
5

Then multiply the intersecting cells by the matching row and column:

3x −1
x 3x2 −1
5 15x −5

We can then combine like terms (in bold) below:

3x −1
x 3x2 −x
5 15x −5

Once we combine our like terms we have our solution: 3x2 + 14x - 5.

Example 2 - Using the Punnett Square

Let's say we wanted to multiply a binomial by a trinomial:

(7x + 2)(4x2 - 3x + 9) =

We could do this by making a Punnett Rectangle with two rows and three columns (or two columns and three rows):

4x2 −3x 9
7x
2

We would then multiply the corresponding cells together:

4x2 −3x 9
7x 28x2 −21x 63x
2 8x2 −6x 18

We then want to combine like terms (in bold and different colors):

4x2 −3x 9
7x 28x3 −21x2 63x
2 8x2 −6x 18

This yields: 28x3 - 13x2 + 57x + 18.

Example 3 - Using the Punnett Square to Multiply Polynomials

Let's now multiply two trinomials:

(4x3 - 2x + 1)(x2 - 5x + 6) =

We'll first create our Punnett Rectangle:

4x3 −2x 1
x2
−5x
6

And then multiply the cells together:

4x3 −2x 1
x2 4x5 −2x3 x2
−5x −20x4 10x2 −5x
6 24x3 −12x 6

And then combine the like terms (in bold):

4x3 −2x 1
x2 4x5 −2x3 x2
−5x −20x4 10x2 −5x
6 24x3 −12x 6

Combining like terms we get our solution: 4x5 - 20x4 + 22x3 + 11x2 - 17x + 6.

Video on Multiplying Polynomials Using the Punnett Square

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