6.2.2 Multiplying Binomials

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Multiplying Binomials Using the Distributive Property

Just like there are different ways to represent multiplication of numbers, there are several different ways to represent how to multiply a binomial times a binomial using the Distributive Property of Multiplication over Addition.

Distribute to multiply the binomials.

Example 1

$(y+5)(y+8)$

Step 1 - Distribute the binomial $(y+8)$.

$y(y+8)+5(y+8)$

Step 2 - Now, distribute the individual terms.

$y^2+8y+5y+40$

Step 3 - Combine like terms.

$y^2+13y+40$

Example 2

$(4y^2+3)(2y^2-5)$

Step 1 - Distribute the binomial $(2y^2-5)$.

$4y^2(2y^2-5)+3(2y^2-5)$

Step 2 - Distribute the individual terms.

$8y^2-20y^2+6y^2-15$

Step 3 - Combine like terms.

$8y^4-14y^2-15$

Multiplying Binomials Using the FOIL Method

If you multiply binomials often enough, you may notice a pattern. Notice that the first term in the result is the product of the first terms in each binomial. The second and third terms are the product of multiplying the two outer terms and then the two inner terms. And the last term results from multiplying the two last terms.

We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for “First, Outer, Inner, Last.” We use this as another way to multiply binomials. The word FOIL is easy to remember and ensures we find all four products.

Let’s multiply $(x+3)(x+7)$ using both methods.

We summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!

How to use the FOIL method to multiply two binomials:

Step 1 - Multiply the First terms.

Step 2 - Multiply the Outer terms.

Step 3 - Multiply the Inner terms.

Step 4 - Multiply the Last terms.

Step 5 - Combine like terms, when possible.

When you multiply by the FOIL method, drawing the arrows will help your brain focus on the pattern and make it easier to apply.

Now we will review examples where we use the FOIL pattern to multiply two binomials.

Multiply the binomials using the FOIL method.

Example 3

$(y-7)(y+4)$

Step 1 - Multiply the $First$ terms.

Step 2 - Multiply the $Outer$ terms.

Step 3 - Multiply the $Inner$ terms.

Step 4 - Multiply the $Last$ terms.

Step 5 - Combine like terms.

$y^2-3y-28$

Example 4

$(4x+3)(2x-5)$

Step 1 - Multiply the First terms, $4x·2x$ .

Step 2 - Multiply the Outer terms, $4x·(-5)$ .

Step 3 - Multiply the Inner terms, $3·2x$ .

Step 4 - Multiply the Last terms, $3·(-5)$ .

Step 5 - Combine like terms.

$8x^2-14x-15$

The FOIL method is usually the quickest method for multiplying two binomials, but it only works for binomials. You can use the Distributive Property to find the product of any two polynomials.

Multiplying Binomials Using Vertical Alignment

Another representation that works for all polynomials is Vertical Alignment. It is much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.

$$\begin{gathered} \underset{―}{\underset{―}{23 \\ \times 46} \\ 138\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t} \\ 920\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}} \\ 1058\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t} \end{gathered}$$

Start by multiplying 23 by 6 to get 138.

Next, multiply 23 by 4, lining up the partial product in the correct columns.

Use a 0 as a place holder in the ones column because you will be multiplying by 4 tens.

Finally, add the partial products.

Example 5

Multiply using Vertical Alignment: $(3y-1)(2y-6)$. It does not matter which binomial goes on the top.

Step 1 - Multiply $3y-1$ by $-6$.

Step 2 - Multiply $3y-1$ by $2y$.

Step 3 - Add like terms.

Notice the partial products are the same as the terms in the FOIL method.

$\begin{array}{c}\begin{array}{c}\hfill (3y-1)\\ \hfill \underset{―}{\times (2y-6)}\\ \hfill -18y+6\\ \hfill 6y^2-2y\\ \hfill 6y^2-20y+6\end{array}\hfill \end{array}$

We have now used three ways to multiply binomials. Be sure to practice each process, and try to decide which one you prefer.