Mathematics » Introducing Polynomials » Multiplying Polynomials

Multiplying a Binomial by a Binomial

Multiplying a Binomial by a Binomial

Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial.

Using the Distributive Property

We will start by using the Distributive Property. Recall the examples from the previous lesson.

 Multiplying a Binomial by a Binomial
We distributed the \(p\) to getMultiplying a Binomial by a Binomial
What if we have \(\left(x+7\right)\) instead of \(p\)?

 

Multiplying a Binomial by a Binomial

Multiplying a Binomial by a Binomial
Distribute \(\left(x+7\right)\).Multiplying a Binomial by a Binomial
Distribute again.\({x}^{2}+7x+3x+21\)
Combine like terms.\({x}^{2}+10x+21\)

Notice that before combining like terms, we had four terms. We multiplied the two terms of the first binomial by the two terms of the second binomial—four multiplications.

Be careful to distinguish between a sum and a product.

\(\begin{array}{cccc}\hfill \mathbf{\text{Sum}}\hfill & & & \hfill \mathbf{\text{Product}}\hfill \\ \hfill x+x\hfill & & & \hfill x·x\hfill \\ \hfill 2x\hfill & & & \hfill {x}^{2}\hfill \\ \hfill \text{combine like terms}\hfill & & & \hfill \text{add exponents of like bases}\hfill \end{array}\)

Example

Multiply: \(\left(x+6\right)\left(x+8\right).\)

Solution

 \(\left(x+6\right)\left(x+8\right)\)
 Multiplying a Binomial by a Binomial
Distribute \(\left(x+8\right)\).Multiplying a Binomial by a Binomial
Distribute again.\({x}^{2}+8x+6x+48\)
Simplify.\({x}^{2}+14x+48\)

Now we’ll see how to multiply binomials where the variable has a coefficient.

Example

Multiply: \(\left(2x+9\right)\left(3x+4\right).\)

Solution

 \(\left(2x+9\right)\left(3x+4\right)\)
Distribute. \(\left(3x+4\right)\)Multiplying a Binomial by a Binomial
Distribute again.\(6{x}^{2}+8x+27x+36\)
Simplify.\(6{x}^{2}+35x+36\)

In the previous examples, the binomials were sums. When there are differences, we pay special attention to make sure the signs of the product are correct.

Example

Multiply: \(\left(4y+3\right)\left(6y-5\right).\)

Solution

 \(\left(4y+3\right)\left(6y-5\right)\)
Distribute.Multiplying a Binomial by a Binomial
Distribute again.\(24{y}^{2}-20y+18y-15\)
Simplify.\(24{y}^{2}-2y-15\)

Up to this point, the product of two binomials has been a trinomial. This is not always the case.

Example

Multiply: \(\left(x+2\right)\left(x-y\right).\)

Solution

 Multiplying a Binomial by a Binomial
Distribute.Multiplying a Binomial by a Binomial
Distribute again.Multiplying a Binomial by a Binomial
Simplify.There are no like terms to combine.

Using the FOIL Method

Remember that when you multiply a binomial by a binomial you get four terms. Sometimes you can combine like terms to get a trinomial, but sometimes there are no like terms to combine. Let’s look at the last example again and pay particular attention to how we got the four terms.

\(\left(x+2\right)\left(x-y\right)\)

\({x}^{2}-\mathit{\text{xy}}+2x-2y\)

Where did the first term, \({x}^{2},\) come from?

It is the product of \(x\phantom{\rule{0.2em}{0ex}}\mathit{\text{and}}\phantom{\rule{0.2em}{0ex}}x,\) the first terms in \(\left(x+2\right)\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\left(x-y\right).\)

Multiplying a Binomial by a Binomial

The next term, \(-\mathit{\text{xy}},\) is the product of \(x\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-y,\) the two outer terms.

Multiplying a Binomial by a Binomial

The third term, \(+2x,\) is the product of \(2\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x,\) the two inner terms.

Multiplying a Binomial by a Binomial

And the last term, \(-2y,\) came from multiplying the two last terms.

Multiplying a Binomial by a Binomial

We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for ‘First, Outer, Inner, Last’. The word FOIL is easy to remember and ensures we find all four products. We might say we use the FOIL method to multiply two binomials.

Multiplying a Binomial by a Binomial

Let’s look at \(\left(x+3\right)\left(x+7\right)\) again. Now we will work through an example where we use the FOIL pattern to multiply two binomials.

Multiplying a Binomial by a Binomial

Example

Multiply using the FOIL method: \(\left(x+6\right)\left(x+9\right).\)

Solution

Step 1: Multiply the First terms.Multiplying a Binomial by a Binomial
Step 2: Multiply the Outer terms.Multiplying a Binomial by a Binomial
Step 3: Multiply the Inner terms.Multiplying a Binomial by a Binomial
Step 4: Multiply the Last terms.Multiplying a Binomial by a Binomial
Step 5: Combine like terms, when possible.Multiplying a Binomial by a Binomial

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 for multiplying two binomials.

  1. Multiply the First terms.
  2. Multiply the Outer terms.
  3. Multiply the Inner terms.
  4. Multiply the Last terms.
  5. Combine like terms, when possible.

Multiplying a Binomial by a Binomial

Example

Multiply: \(\left(y-8\right)\left(y+6\right).\)

Solution

Step 1: Multiply the First terms.Multiplying a Binomial by a Binomial
Step 2: Multiply the Outer terms.Multiplying a Binomial by a Binomial
Step 3: Multiply the Inner terms.Multiplying a Binomial by a Binomial
Step 4: Multiply the Last terms.Multiplying a Binomial by a Binomial
Step 5: Combine like termsMultiplying a Binomial by a Binomial

Example

Multiply: \(\left(2a+3\right)\left(3a-1\right).\)

Solution

 Multiplying a Binomial by a Binomial
 Multiplying a Binomial by a Binomial
Multiply the First terms.Multiplying a Binomial by a Binomial
Multiply the Outer terms.Multiplying a Binomial by a Binomial
Multiply the Inner terms.Multiplying a Binomial by a Binomial
Multiply the Last terms.Multiplying a Binomial by a Binomial
Combine like terms.Multiplying a Binomial by a Binomial

Example

Multiply: \(\left(5x-y\right)\left(2x-7\right).\)

Solution

 Multiplying a Binomial by a Binomial
 Multiplying a Binomial by a Binomial
Multiply the First terms.Multiplying a Binomial by a Binomial
Multiply the Outer terms.Multiplying a Binomial by a Binomial
Multiply the Inner terms.Multiplying a Binomial by a Binomial
Multiply the Last terms.Multiplying a Binomial by a Binomial
Combine like terms. There are none.Multiplying a Binomial by a Binomial

Using the Vertical Method

The FOIL method is usually the quickest method for multiplying two binomials, but it works only for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.

Multiplying a Binomial by a Binomial

You start by multiplying \(23\) by \(6\) to get \(138.\)

Then you multiply \(23\) by \(4,\) lining up the partial product in the correct columns.

Last, you add the partial products.

Now we’ll apply this same method to multiply two binomials.

Example

Multiply using the vertical method: \(\left(5x-1\right)\left(2x-7\right).\)

Solution

It does not matter which binomial goes on the top. Line up the columns when you multiply as we did when we multiplied \(23\left(46\right).\)

 Multiplying a Binomial by a Binomial
Multiply \(2x-7\) by \(-1\).Multiplying a Binomial by a Binomial
Multiply \(2x-7\) by \(5x\).Multiplying a Binomial by a Binomial
Add like terms.Multiplying a Binomial by a Binomial

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

Multiplying a Binomial by a Binomial

We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The three methods are listed here to help you remember them.

Definition: Multiplying Two Binomials

To multiply binomials, use the:

  • Distributive Property
  • FOIL Method
  • Vertical Method

Remember, FOIL only works when multiplying two binomials.

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