FOIL
One of the most used skills you need moving forward in math courses is the ability to factor quadratic expressions. You will have to be as quick with it as you should be with basic multiplication facts. This short-cut for binomials will be a life-saver with this required skill.
You will need to be quick and accurate with this short-cut, but to get there takes time, patience, reflection and understanding. Use the tabs below to navigate this important information, the examples, and so on. Work through the first section, watch the video, try the practice problems. Then, work through the second section, watch the video and try the practice problems. Do not allow yourself to get rusty with this skill, it is one that will pay off for you, or punish you should you forget it!
There’s a pattern to multiplying binomials that is often referred to as FOIL. We did not teach this short-cut initially because it only works for a pair of binomials being multiplied. The philosophy is to learn what always works first, then tricks and short-cuts as they come up.
Let’s take a look at multiplying two binomials, the official way, so we can see how FOIL works.
(x + 1)(2x – 5)
This of course has a meaning. The first binomial is being distributed to the second. If we set this up, we get the following:
x(2x – 5) + 1(2x – 5)
Here we can already see FOIL. FOIL stands for First, Outer, Inner, Last. When we multiply the x and the 2x, that is multiplying the first part of each binomial. When we multiply the x and the negative five, that is the physical outside of the entire original expression. Next, you’d multiply 1 and 2x, and in the original expression those are the inner terms. When the 1 and -5 are multiplied, those are the last two terms of each binomial.
What is typically the case is that the product of the outer (O) and inner (I) are like terms and they can be combined. We’ll call that the OI, for now.
Let’s see an example.
(2x + 4)(x + 9)
Here the First is 2x and x, which has the product of 2x2. The Outer is 2x and 9, which makes 18x. The Inner is 4 and x, which is 4x, and the Last is 4 and 9, which makes 36.
With a little practice this becomes something that can be done almost entirely in your head. This is an important skill to have as efficiency and accuracy will be crucial moving forward. Soon you will be factoring (un-multiplying) these types of products. In order to do so, you must be proficient with multiplication.
At this point you have the idea, but need some practice. Don’t read any further, yet. Go to the assignment and complete part 1. Take your time, this is an important skill. Follow the directions for each section.
Now that you have some practice under your belt, let’s see a few examples and look at some important patterns. As you look at this table below, think about the original expressions (the problems) and what is different about each.
Did you notice that the problems each have the same combination of numbers, 1 and 8, just with different signs?
Let’s develop some language to discuss each term in the answer. These answers are all of degree 2 (biggest exponent of two), and that makes they quadratic. (You might remember that from the study you did on functions.) The standard form of a quadratic is as follows.
ax2 + bx + c
The coefficients are a, b, and c. The leading coefficient is a, and it will never be zero. If it was, we wouldn’t have a quadratic expression anymore. The constant term is c. The coefficient of x1 is b. So, in the expression 4x2 – 3x – 2, a is 4, b is – 3, and c is – 2. In x2 – x, a is 1, b is – 1, and c = 0. In x2 – 25, a = 1, b = 0, and c = -25.
Notice that in our table above, all of the first terms are both x, and the products all have an a of 1. The leading coefficient will only be 1 if both First terms are x.
Now, look at the sign of the c terms in the answers. See if you can figure out a relationship between the sign of c and the factors (parts that multiply together to make the product).
Did you notice that if c is negative, the Last terms of the factors have different signs? If the c is positive, the Last terms of the factors are the same sign! What about the b term? Do you see a pattern there?
- If a = 1, then the First terms in the factors are both x*.
- It could be a combination of say 3 and 1/3, but usually just 1x and 1x.
- If c is positive, the Last terms of each factor are the same sign.
- If c is positive and b is positive, then both Last terms in the factors are positive.
- If c is positive and b is negative, both Last terms in the factors are negative.
- If c is negative, the Last terms of each factor are different signs.
- If c is negative and b is positive, the larger Last term is positive.
- If c is negative and b is negative, the larger Last term is negative.
You really need to really know and understand this, without much thinking and without having to look it up. It needs to be an automatic understanding! Keep it in mind as you continue to learn how FOIL works.
Let’s talk about the most common mistake in FOIL…the perfect square!
(x + 4)2
Students always say this x2 + 16, but it is not! Remember, an exponent of two means that the base is multiplied by itself. Here the base is a binomial. A binomial multiplied by itself is set up as follows.
(x + 4)(x + 4)
If you carry out the distribution here, by FOIL or long-hand, it is x2 + 8x + 16. This is a perfect square, or a square number (polynomial). A square number is a number that is the product of another number that has been squared. For example, 25 is a square number because 52 = 25.
The trinomial x2 + 10x + 25 is a perfect square also because (x + 5)2 = x2 + 10x + 25.
If a quadratic is a perfect square (when a = 1), notice that if you take half of b, then square it, you arrive at the value of c. Half of 10 is 5, and 5 × 5 = 25.
This is another one of those patterns you must recognize without hesitation or failure. Keep it in mind as you work through these lessons.
There are two things to be cautious of here. First, how you square a binomial is important. You cannot distribute an exponent over addition. That would be a violation of the order of operations. When you see something like (A + B)2, it means (A + B)(A + B), not A2 + B2.
The second thing to be aware of is the pattern of the squared binomial. Half of the middle coefficient (b) is equal to the last term (c). For example, x2 – 18x + 81. Half of b is – 9, and (-9)2 = 81, which is the c term.
The last special issue we will discuss is something called difference of squares. That is a binomial where the first and second terms are square numbers, and they’re subtracting. An example is 9x2 – 16. Since 9x2 is a perfect square, (3x)2, and 16 is a perfect square, (4)2, and they’re subtracting, this is called difference of squares.
There’s a pattern of binomial multiplication that produces these difference of squares. (3x + 4)(3x – 4) would yield this difference of squares. Essentially if the first terms of each binomial are the same, and the second terms are opposite numbers (same absolute value, different signs), then the product of those binomials will be a difference of squares. For example, (x – 6)(x + 6) gives us x2 – 36.
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