Although factoring is a fundamental skill of high school level math, many students experience difficulties. If this includes you, keep reading to become an expert at factoring!
The Golden Rule: Common Factor First
Common factoring is the process of finding numbers and/or variables that are a multiple of every term in an expression and removing them. For example, 2 and x are both common factors of the expression 2x^3 + 8x^2 + 12x. Once these have been identified, we divide them out of the expression and determine what remains. Keeping with the above example, factoring out 2 and x would give 2x(x^2 + 4x + 6). By “pulling out” the common factors, the remaining expression becomes simpler and easier to handle.
The Basic Approach to Quadratics: Decomposition
Quadratics are the most common expression that high school students factor. Decomposition is a relatively simple and full-proof approach to these problems. Just follow these steps:
1) Identify the a, b, and c values. For those who need a refresher ax^2 + bx + c is the standard form of a quadratic.
for example, with 4x^2 – 12x + 5
a = 4, b = -12, c = 5
2) Find two numbers that multiply to the product of ac and add to the value of b.
for example, with 4x^2 – 12x + 5
the numbers are -2 and -10 since (-2) + (-10) = (-12) and (-2)(-10) = 20
3) Rewrite the quadratic with your numbers instead of b (this is the decomposition part).
for example, 4x^2 – 12x + 5 becomes
4x^2 -2x -10x + 5 or
4x^2 -10x -2x + 5
4) Common factor the the halves of the expressions separately.
for example, with 4x^2 -2x -10x + 5, it becomes
2x(2x-1) -5(2x-1)
with 4x^2 -10x -2x + 5
it becomes 2x(2x-5) – (2x-5)
5) Common factor step 4’s expression
for exmaple, with 2x(2x-1)-5(2x-1)
the answer is (2x-1) (2x-5)
with 2x(2x-5) – (2x-5)
the answer is (2x-5) (2x-1)
Shortcuts:
Luckily some shortcuts exist to make life easier!
Simple Trinomials:
If the a value is 1, this is called a simple trinomial. Using the numbers from step 2), skip to the answer, inserting these numbers straight into the brackets with x.
for example, with x^2 +3x +2
a = 1
the numbers are 1 and 2 since 1+2 = 3 and (1)(2) = 2
so the answer is (x+1)(x+2)
Difference of Squares:
If the b value is 0, c is negative, and the square root of the absolute value of c is a whole number, then this is a difference of squares. The numbers are the positive and negative roots of the absolute value of c. Again, skip to the answer and insert these numbers straight into the brackets with x.
for example, with x^2 – 4
since the square root of 4 is 2,
the answer is (x-2)(x+2)
Pro Tips:
If you are having difficulty with step 4), try using the other version of the rewritten quadratic.
Check your answers by expanding!
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