How to Factor Variables – Common Terms
In order to know how to factor variables, you need to find out what the a,b, or c terms in an expression have in common.
Exercises on How to Factor Variables
Instructions: Factor out the greatest common factor in each of the following expressions. You may want to view the examples below the quiz first.
[WpProQuiz 15]
How to Factor Out Variables
Variables in an equation represent unknown values.
Look at the variables in the following expression:
ab + bc – bd
When we factor out variables, we need to look to see which variables are common to all of the terms in the expression.
In out expression above, ab is the first term, bc is the second term, and bd is the third term.
We can see that all of these three terms have got the variable b in them.
So, we can factor out the variable as shown below:
First, show the multiplication for each term.
ab + bc – bd =
(a × b) + (b × c) – (b × d)
Then change the positions of the variables inside each set or parentheses to put the common variable first.
(a × b) + (b × c) – (b × d) =
(b × a) + (b × c) – (b × d)
Then remove the b variable from each set of parenthesis and place it at the front of your new expression.
You can cancel out the common variable as you solve the problem.
Be careful with the plus and minus signs when you do this.
(b × a) + (b × c) – (b × d) =
b[(b × a) + (b × c) – (b × d)] =
b(a + c – d)
Factoring Variables with Exponents
In the section above, we have shown a very basic example on how to factor out variables from an expression or equation.
On the exam, “factoring variables” problems will be much more difficult than this.
Factoring problems on the test will usually involve terms that have both numbers and variables, and the variables will often have exponents.
So, you will need to know exponent laws to solve these types of problems.
Look at the example factoring variables question below and study the steps that follow.
abc + a3b2c2 – a2b3 + ab4d
First of all, show the multiplication for each term.
abc + a3b2c2 – a2b3 + ab4d =
(a × b × c) + (a3 × b2 × c2) – (a2 × b3) + (a × b4 × d)
Then show the multiplication for each exponent. Remember that the exponent indicates how many times to multiply that variable by itself.
(a × b × c) + (a3 × b2 × c2) – (a2 × b3) + (a × b4 × d) =
(a × b × c) + [(a × a × a) × (b × b) × (c × c)] – [(a × a) × (b × b × b)] + [a × (b × b × b × b) × d]
Next look for common factors. Here, we can see that each term has one a and one b in common.
So, cancel out one a and one b from each term.
(a × b × c) + [(a × a × a) × (b × b) × (c × c)] – [(a × a) × (b × b × b)] + [a × (b × b × b × b) × d] =
ab{(a × b × c) + [(a × a × a) × (b × b) × (c × c)] – [(a × a) × (b × b × b)] + [a × (b × b × b × b) × d]} =
ab{(c) + [(a × a) × (b) × (c × c)] – [(a) × (b × b)] + [(b × b × b) × d]}
Then, express the result with exponents.
ab{(c) + [(a × a) × (b) × (c × c)] – [(a) × (b × b)] + [(b × b × b) × d]} =
ab{(c) + [(a2) × (b) × (c2)] – [(a) × (b2)] + [(b3) × d]}
Finally, express each of the sets of multiplication as individual terms.
ab{(c) + [(a2) × (b) × (c2)] – [(a) × (b2)] + [(b3) × d]} =
ab[c + (a2bc2) – (ab2) + (b3d)] =
ab(c + a2bc2 – ab2 + b3d)
Further information on factoring variables
You can perform distribution to check your factoring.
In addition to factoring variables, you will need to know how to factor out numbers.
You should also look at our posts on factoring expressions.
