[MUSIC PLAYING] Let's look at our objectives for today. We'll start by reviewing the distributive property. We'll then introduce binomial multiplication.
We'll then introduce an acronym, FOIL, used for binomial multiplication. We'll do some examples multiplying binomials with FOIL. And finally, we'll look at FOIL with the binomial squared.
Let's start by reviewing the distributive property. For example, we wanted to simplify this expression. 8x squared times 2x to the third minus 1. We want to distribute the 8x squared to both terms in the parentheses. So to multiply 8x squared by 2x to the third, we multiply 8 times 2, which gives us 16, and x squared times x to the third, which gives us x to the fifth. Because the bases are both x, we can add our exponents.
We then multiply 8x squared by negative 1, which gives us a negative 8x squared. So our final answer is 16x to the fifth minus 8x squared. And our answer is in standard form or descending order according to the exponents in each term.
Now, let's do an example multiplying binomials. We want to multiply 3x plus 3 times 2x minus 4. To multiply binomials, we need to distribute twice multiplying the 3x by both terms in the second parentheses and then the 3 by both terms in the second parentheses. When distributing, we will write all terms as a single expression-- adding terms with positive coefficients and subtracting terms with negative coefficients.
So let's see with that looks like. 3x and 2x both have implied exponents of 1, so 3 times 2 is 6 and x times x is x squared. 3x times 2x gives us 6x squared. We then multiply 3x times negative 4, which gives us negative 12x. Then we multiply 3 by 2x, which gives us 6x, and 3 by negative 4, which gives us negative 12. We can combine the like terms, negative 12x and 6x, which gives us negative 6x. So our final answer is 6x squared minus 6x minus 12.
Here is a second example. 5x minus 3 times x plus 2. 5x times x will give us 5x squared. 5x times 2 is 10x. Then negative 3 times x is negative 3x, and negative 3 times 2 is negative 6. We can combine the like terms, 10x and negative 3x, to give us 7x. So our final answer is 5x squared plus 7x minus 6.
Now, let's look at that last example again to see how we can multiply binomials using an acronym called "FOIL." In our last example, we first distributed multiplying the first two terms in each parentheses and then the outside terms. We then distributed multiplying the inside terms and then the last two terms in each parentheses. So one way to remember how to multiply binomials is using the acronym FOIL. FOIL is an acronym to remember the steps for distributing factors in binomial multiplication where FOIL stands for "First, Outside, Inside, and Last."
Here is another example. Suppose a farmer wants to plant a small area for a new chicken pen. The length and width of the pen are shown below. What is the area of the chicken pen in terms of x?
To find the area of the pen, we want to multiply the length and the width. This will give us x plus 3 times x plus 10. These are binomials multiplied together, so we can multiply using FOIL. Our first two terms, x and x, multiply together to give us x squared.
Our outside terms, x times 10, will give us 10x. Multiplying our inside terms, 3 and x gives us 3x. And multiplying our last terms, 3 and 10 gives us 30. We can combine our like terms, 10x and 3x, giving us a final answer of x squared plus 13x plus 30. So the area of the chicken pen can be written as x squared plus 13x plus 30.
Here is another example. We want to multiply 2x squared plus 4x times 3x plus 2. So we start by multiplying our first terms together. 2x squared and 3x gives us 6x to the third. We then multiply our outside terms. 2x squared times 2 gives us 4x squared.
Multiplying our inside terms, 4x times 3x gives us 12x squared. And multiplying our last terms, 4x times 2 gives us 8x. 4x squared 12x squared are like terms, so combining gives us 16x squared. So our final expression is 6x to the third plus 16x squared plus 8x.
Here is our last example. We want to simplify x minus 5 squared. This is an example of a binomial squared. x minus 5 squared means x minus 5, x minus 5. So we can multiply using FOIL. Multiplying our first two terms, x times x gives us x squared.
Multiplying our outside terms, x times negative 5 is negative 5x. Our inside terms, negative 5 times x is negative 5x. And our last terms, negative 5 times negative 5 is a positive 25. We can combine our like terms, negative 5x and negative 5x, which gives us a negative 10x. So our final expression is x squared minus 10x plus 25.
Let's go over our important points from today. Make sure you get these in your notes so you can refer to them later. FOIL is an acronym used to remember the steps for distributing factors in binomial multiplication-- First, Outside, Inside, and Last. In standard form, the terms of a polynomial are in descending order according to the exponents in each term. When multiplying monomials, remember to use the product property of exponential expressions to add the exponents when the bases are the same.
So I hope that these important points and examples helped you understand a little bit more about multiplying binomials. Keep using your notes, and keep on practicing. And soon, you'll be a pro. Thanks for watching.
An acronym to remember the steps for distributing factors in binomial multiplication: first, outside, inside, last.