High School: Algebra
Seeing Structure in Expressions HSA-SSE.A.2
2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).
In English, we have many different ways of saying the same thing. "Stop that!" and "Stop eating that!" and "Stop eating my sandwich!" all mean the same thing, but it's important to be able to know which one to use. Well, if your sandwich is being eaten, then any of those will probably work.
Math works much the same way. Writing mathematical expressions in different ways is incredibly important, especially in algebra. It's not about redundancy; it's about simplicity. And about keeping a firm grip on that sandwich of yours.
Students should be able to convert mathematical expressions to alternative but equivalent forms by factoring. This is important when students want to explain certain properties of an expression or the quantity which the expression represents, solve equations involving mathematical expressions, or simplify complex expressions. They might not want to at first, but if you tell them it's on the chapter test, they'll most likely be very interested.
Factoring is the conversion of a mathematical expression into an equivalent form that consists of a single term composed of nothing but terms multiplied together, called factors. Basically, it's the opposite of the distributive property. Students should recall that the distributive property is:
a(b + c) = ab + ac
Note that the a has been distributed to the two terms inside the parentheses. Also note that we have taken an expression with one term and transformed it into an equivalent expression with more than one term. We've expanded the expression.
Factoring is the reverse process, so it can be stated as:
ab + ac = a(b + c)
We have converted from a two-term expression to a one-term expression which consists of nothing but factors, namely a and (b + c). We repeat the process on any factors which can be factored further. Once we have factored everything which can be factored, the expression is said to be fully factored. Fancy that.
The factor we pull outside the parenthesis (a, in our previous example) is called the greatest common factor or GCF if it is the largest factor that is common to all of the terms. The GCF can be a number, a variable, a multi-term expression, or any combination of these.
The GCF technique is the very first thing that should be checked when confronted with any factoring problem. Doing so will often simplify subsequent factoring steps. Strong multiplication and division skills are critical to being efficient at GCF determination, so it may be helpful to go over divisibility rules with your students. (All even numbers are divisible by 2, if the sum of a number's digits is divisible by 3 then the number itself is divisible by 3, and so on.)
Factoring by grouping is a variant of the GCF technique. In this case, we have to group certain terms together, then treat each grouping as a GCF problem. We continue that process until we just can't go any further.
For instance, we can factor the expression 8x2 – 4x – 40x + 20 by a common numerical factor of 4. There is no common variable factor, though. We can factor this as 4(2x2 – x – 10x + 5). What about that mess inside the parentheses?
If we group the first two terms together, we see they have a common factor of x, so they can be rewritten as x(2x – 1). The last two terms have a common factor of -5. They can then be written as -5(2x – 1). Putting this all together we get the following:
4[x(2x – 1) – 5(2x – 1)]
Our four-term expression is now a two-term expression. These two terms have a common factor of (2x – 1), however, so we can factor further:
4(2x – 1)(x – 5)
This is the fully factored form.
As important as GCF factorizations are, students should know how to factor expressions of a higher order. Yes, we're talking about exponents.
Students should know that expressions of order two are called quadratic expressions, and that they can be written in the form ax2 + bx + c, where a, b, and c are numbers and a ≠ 0. (Ask the students to explain why that restriction is necessary.)
Factoring quadratics is a little more challenging, so it helps to examine the factoring process in reverse. Consider the expression (2x – 5)(3x + 2). Clearly, this is a factored form of a quadratic expression. If we use the distributive property to expand the expression we would get
(2x – 5)(3x + 2)
2x(3x + 2) – 5(3x + 2)
6x2 + 4x – 15x – 10
6x2 – 11x – 10
The last expression is the quadratic expression written in standard form. We can identify a = 6, b = -11, and c = -10.
Notice that if we run this distributive process in reverse, it's like taking the standard form quadratic and splitting up the middle term into two pieces and then using the grouping technique. How would we know to break up the middle term in that way, though? There are many ways to write -11 as the sum of two numbers, but only one will give us what we need.
How would we know to pick +4 and -15? Notice that the product of +4 and -15 is -60. This is also the product of the first and last coefficients in the standard form, a and c: (6)(-10) = -60. So, one might assume we multiplied a and c together, then found two numbers which multiply together to get that number but add together to give b. If you study several examples, you'll see this is correct. This is always the case. So the steps in the general technique for factoring quadratics which can be factored are:
- Write the quadratic in standard form.
- Multiply a and c together.
- Find two numbers which multiply to produce the number in step 2, but add to produce b. Be careful with signs!
- Rewrite the quadratic, but replace the middle term with two terms using the two numbers from step 3.
- Use the grouping technique to factor the expression.
Students should know that this general technique works for all factorable quadratics, but that there are some special cases to consider. For instance, the difference of two squares x2 – y2 can be factored as (x – y)(x + y). Also, the perfect trinomial x2 + 2ax + a2 can be factored to equal (x + a)2. These two work even when x, y, and a are quantities made of multiple terms. That's pretty nifty.
In general, students should follow these rules:
- Determine if there is a greatest common factor in the expression. You typically want to factor that out first, as it leaves a simpler expression to concentrate on in subsequent steps.
- Determine if it is possible to use the grouping technique.
- If the expression is a quadratic, put it in standard form first.
- Examine the form of the expression and determine if one of the special forms—the difference of two squares or the perfect square trinomial—applies. If so, apply the rules given previously for writing the factored form.
- If the quadratic is not a special form, use the general method for factoring quadratics.
Factoring takes a lot of practice, but it is probably the single most important skill students will develop in math. It is critical to just about every topic in higher levels of math, so it is worthwhile to spend a fair amount of time developing this skill. Once students start getting the hang of it, many of them will be able to just "see" the factors. Encourage them to always look for patterns, to pay attention to the individual pieces of the expressions they get and how they appear to relate to the pieces of the original expression.
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