When a letter like a, b, x or y pops up in a mathematical expression, it's called a variable, but really it's a placeholder that represents a number of unknown value. You can perform all the same mathematical operations on a variable that you'd perform on a known number. That fact comes in handy if the variable pops up in a fraction, where you'll need tools like multiplication, division and canceling of common factors to simplify the fraction.
Combine Like Terms
Factor and Cancel
Factor Into a Mixed Number
Use Standard Formulas to Factor
The standard formula for the difference of squares is:
(x2 - y2) = (x - y)(x + y)
Combine like terms in both the numerator and the denominator of the fraction. When you first start handling fractions with variable, this may be done for you. But later on, you might encounter "messier" fractions like the following:
(a + a) / (2_a_ - a)
When you combine like terms, you end up with a much more civilized fraction:
Factor the variable out of both numerator and denominator of the fraction if you can. If the variable is a factor in both places, you can then cancel it. Consider the simplified fraction just given:
As a quick aside, any time you see a variable by itself, it's understood to have a coefficient of 1. So this could also be written as:
Which makes it more obvious that when you cancel the common factor a from both the numerator and denominator of the fraction, you're left with the following:
Which, in turn, simplifies to the whole number 2.
What if you have a fraction like 3_a_/2? You can't factor a out of both the numerator and the denominator of the fraction, but because it's in the numerator, you can treat it as a whole number. To make sense of this, first write the fraction out thusly:
You can insert the 1 in the denominator thanks to the multiplicative identity property, which states that when you multiply any number by 1, the result will be the original number you started with. So you haven't changed the value of the fraction at all; you've just written it a little differently.
Next, separate the factors thusly:
a/1 × 3/2
And simplify a/1 to a. This gives you:
a × 3/2
Which can be simply written as the mixed number:
What if you end up with a messy fraction like the following?
(b2 - 9) / (b + 3)
At first glance there's no easy way to factor b out of both numerator and denominator. Yes, b is present in both places, but you'd have to factor it out of the entire term in both places, which would give you the even messier b(b - 9/b) in the numerator and b(1 + 3/b) in the denominator. That's a dead end.
But if you've been paying attention in your other lessons, you might notice that the numerator can actually be rewritten as (b2 - 32), also known as "the difference of squares," because you're subtracting one squared number from another squared number. And there's a special formula that you can memorize to factor the difference of squares. Using that formula, you can rewrite the numerator as follows:
(b - 3)(b + 3)
Now, take a look at that in the context of the entire fraction:
(b - 3)(b + 3) / (b + 3)
Thanks to that standard formula you either memorized or looked up, you now have the identical factor (b + 3) in both the numerator and the denominator of your fraction. Once you cancel that factor, you're left with the following fraction:
(b - 3) / 1
Which simplifies to just:
(b - 3)