Multiplying Terms Using the FOIL Method

Hello, and welcome to this video about multiplying terms using the FOIL method. Today we’ll learn a strategy for multiplying two-digit numbers. We’ll also learn how to apply this strategy to multiplying polynomials with two terms.

Before we get started, let’s review a few things. First, a polynomial is an expression that can have many terms with variables, coefficients, and positive exponents. These terms can be added, subtracted, multiplied, and divided.

Today we’re going to talk about polynomials with two terms. Polynomials with two terms are called binomials.

Examples of binomials are $$2x+7, 30\text{ }+8,\text{ and}\text{ }5x^{2}-3$$. All binomials have two terms that are separated by an operation sign.

The FOIL method is a technique that’s used to remember the steps for multiplying two binomials. The letters in the word FOIL stand for First, Outer, Inner, and Last. The product of two binomials is the sum of multiplying the first terms, outer terms, inner terms, and last terms:

$$(a+b)(c+d)=ac+ad+bc+bd$$

Let’s look at an example of multiplying two-digit numbers using the FOIL method.

So consider the multiplication problem $$35\times 56$$. To solve using the FOIL method, start by writing each term as a binomial. In other words, split each number into two numbers according to their place value. So $$35$$ is $$(30+5)$$ and $$56$$ is $$(50+6)$$.

Now that each number is written as a binomial, we can apply the FOIL method.

So the F in FOIL stands for First. We’re going to multiply our first terms by one another.

$$30\times 50=1,500$$

Then, the O in FOIL stands for Outer, so we’re going to multiply our outer numbers.

$$30\times 6=180$$

Then the I in FOIL stands for Inner, so we’re going to multiply the inner terms.

$$5\times 50=250$$

And finally, we have the L in FOIL, which stands for Last. So the last two terms of our binomials are 5 and 6.

$$5\times 6=30$$

Now, all we have to do is add this together and we’ll get a total of: $$1,500+180+250+30=1,960$$.

Let’s look at another multiplication problem where we can apply the FOIL method.

Makayla has 24 filled photo albums. If each album holds 48 photos, how many photos does she have in all?

To solve this problem, we need to multiply $$24\times 48$$. To solve using the FOIL method, start by writing each term as a binomial, so $$24$$ is $$(20+4)$$ times 48 is $$(40+8)$$.

$$(20+4)(40+8)$$

Now that each number is written as a binomial, we can apply the FOIL method.

The F in FOIL stands for First, so we’ll multiply our first terms by one another.

$$20\times 40=800$$

Then we’ll do our outer terms because that’s what the O stands for.

$$20\times 8=160$$

Next comes I, which stands for Inner.

$$4\times 40=160$$

And finally, we’re going to do the L, which is the last two terms.

$$4\times 8=32$$

Now, all we have to do is add these four numbers together and we get: $$800+160+160+32=1,152$$. So Makayla has 1,152 photos altogether.

The FOIL method is a good strategy for multiplying two-digit numbers, especially if you don’t have a calculator on hand. The FOIL method can also be applied to binomials with variables and exponents. Let’s look at another problem together:

$$(6x+3)(4x+7)$$

Notice that this problem has the variable $$x$$ in the terms $$6x$$ and $$4x$$. Even so, we can apply the FOIL method to multiply.

So we’re going to start by multiplying our first terms together.

$$6x\times 4x=24x^{2}$$

Then we’re going to multiply our outer terms.

$$6x\times 7=42x$$

Next, we’ll do our inner terms, which is $$3$$ and $$4x$$.

$$3\times 4x=12x$$

And finally, we’ll do our last terms.

$$3\times 7=21$$

Now all we have to do is add our like terms right here.

$$24x^{2}+42x+12x+21$$

$$24x^{2}+54x+21$$

So that’s our answer: $$24x^{2}+54x+21$$.

I have one more problem for you to analyze on your own. This one is a little bit more challenging, but I know you can do it.

The length of a rectangle is $$(5x+1)$$. Its width is $$(2x+8)$$. Write a polynomial for the area of the rectangle in standard form, $$ax^{2}+bx+c$$.

Pause the video, analyze the word problem, and see if you can solve it by yourself. When you’re ready, resume the video, and we’ll go over the answer together.

Now that you’ve tried this problem, let’s go over it together.

To find the area of a rectangle, multiply its length by its width.

$$A=lw$$

Since the length is $$(5x+1)$$, we can write that. And the width is $$(2x+8)$$.

$$A=(5x+1)(2x+8)$$

Now we can multiply using the FOIL method.

So the first two terms are $$5x$$ and $$2x$$.

$$5x\times 2x=10x^{2}$$

Plus our outer terms.

$$5x\times 8=40x$$

Plus our inner terms.

$$1\times 2x=2x$$

And finally, our last terms.

$$1\times 8=8$$

$$10×2+40x+2x+8$$

Now all we have to do is combine our like terms.

$$10×2+42x+8$$

Therefore, the area of the rectangle is $$10x^{2}+42x+8$$.

Great job!

I hope this video about multiplying terms using the FOIL method was helpful. Thanks for watching, and happy studying!

Practice Questions

Question #1:

Solve $$48×72$$ using the FOIL method.

$$936$$
$$1{,}264$$
$$3{,}456$$
$$3{,}650$$

The product of two binomials is the sum of multiplying the first terms, outer terms, inner terms, and last terms.

 $$(40+8)(70+2)$$ Before applying the FOIL method, start by writing each term as a binomial. $$40\times70=2{,}800$$ Next, multiply the first terms. $$40\times2=80$$ Then, multiply the outer terms. $$8\times70=560$$ From here, multiply the inner terms. $$8\times2=16$$ After that, multiply the last terms. $$2,800+80+560+16=3{,}456$$ Finally, add the products together to get a total of $$3{,}456$$.
Question #2:

Multiply the binomials using the FOIL method. Write your answer in standard form, $$ax^2+bx+c$$.

$$(7x+8)(2x+1)$$

$$14x^2+9x+9$$
$$32x+8$$
$$9x^2+23x+8$$
$$14x^2+23x+8$$

The product of two binomials is the sum of multiplying the first terms, outer terms, inner terms, and last terms.

 $$(7x+8)(2x+1)$$ $$7x\cdot2x=14x^3$$ Start applying the FOIL method by multiplying the first two terms. $$7x\cdot1=7x$$ Next, multiply the outer terms. $$8\cdot2x=16x$$ Then, multiply the inner terms. $$8\cdot1=8$$ From here, multiply the last terms. $$14x^2+7x+16x+8$$ Create one expression by adding the partial products together. $$14x^2+23x+8$$ Finally, write the expression in standard form by combining like terms.
Question #3:

Multiply the binomials using the FOIL method. Write your answer in standard form, $$ax^2+bx+c$$.

$$(x-4)(2x+9)$$

$$3x^2-x-36$$
$$2x^2+x-36$$
$$2x^2-x+36$$
$$3x^2+x+36$$

The product of two binomials is the sum of multiplying the first terms, outer terms, inner terms, and last terms.

 $$(x-4)(2x+9)$$ $$x\cdot2x=2x^2$$ Start applying the FOIL method by multiplying the first two terms. $$x\cdot9=9x$$ Next, multiply the outer terms. $$-4\cdot2x=-8x$$ Then, multiply the inner terms. $$-4\cdot9=-36$$ From here, multiply the last terms. $$2x^2+9x-8x-36$$ Create one expression by adding the partial products together. $$2x^2+x-36$$ Finally, write the expression in standard form by combining like terms.
Question #4:

Sal is planting a vegetable garden in his backyard. The length of the garden is $$(3x+4)$$. Its width is $$(2x-5)$$. Use the FOIL method to write a polynomial for the area of the garden. Write your answer in standard form, $$ax^2+bx+c$$.

$$5x^2-7x+20$$
$$6x^2+7x-20$$
$$5x^2+7x+20$$
$$6x^2-7x-20$$

To find the area of the garden, multiply the length by the width. Since both measurements are given as binomials, we can apply the FOIL method to solve. The product of two binomials is the sum of multiplying the first terms, outer terms, inner terms, and last terms.

 $$(3x+4)(2x-5)$$ $$3x\cdot2x=6x^2$$ Start applying the FOIL method by multiplying the first two terms. $$3x\cdot(-5)=-15x$$ Next, multiply the outer terms. $$4\cdot2x=8x$$ Then, multiply the inner terms. $$4\cdot(-5)=-20$$ From here, multiply the last terms. $$6x^2-15x+8x-20$$ Create one expression by adding the partial products together. $$6x^2-7x-20$$ Finally, write the expression in standard form by combining like terms.

The area of Sal’s vegetable garden is $$6x^2-7x-20$$.

Question #5:

Shannon is shopping for doormats. The length of her current rectangular doormat is 3 times longer than its width, w. She wants to get a new doormat that is 1.5 feet longer and 1.5 feet wider than her current doormat. Write an expression with two binomials that represents this scenario. Then, multiply the binomials using the FOIL method to write a polynomial for the area of the new doormat. Write your answer in standard form, $$ax^2+bx+c$$.

$$3w^2+60w+225$$
$$3w^2+6w+2.25$$
$$4w^2+3w+3$$
$$4w^2+3w+2.25$$

To find the area of the new doormat, multiply the length by the width. Since the new doormat’s width is 1.5 feet wider than the old doormat, the width is $$(w+1.5)$$. Since the new doormat’s length is 3 times the original width and 1.5 feet longer than the old doormat, the length is $$(3w+1.5)$$.

$$(w+1.5)(3w+1.5)$$

Since both measurements are given as binomials, we can apply the FOIL method to solve. The product of two binomials is the sum of multiplying the first terms, outer terms, inner terms, and last terms.

 $$w\cdot3w=3w^2$$ Start applying the FOIL method by multiplying the first two terms. $$w\cdot1.5=1.5w$$ Next, multiply the outer terms. $$1.5\cdot3w=4.5w$$ Then, multiply the inner terms. $$1.5\cdot1.5=2.25$$ From here, multiply the last terms. $$3w^2+1.5w+4.5w+2.25$$ Create one expression by adding the partial products together. $$3w^2+6w+2.25$$ Finally, write the expression in standard form by combining like terms.

The area of Shannon’s new doormat is $$3w^2+6w+2.25\text{ feet}$$