First, we answer the question, “What is factoring used for?”
Students need to be given a reason for studying. The answer here is this is a powerful tool that can be used in equation solving.
In factoring we break down expressions to two or more expressions that are multiplied by each other. The parts used in that multiplication are called factors.
First, remove any common factor.
Factor 5x3 + 5x2 + 25x. each term contains 5x, so it can be removed as a common factor.
5x(x2 + x + 5)
It is important that common factors be removed first, since doing this makes the problem easier.
Factoring by grouping is a form of removing common factors.
Factor x3 – x2 + 7x – 7
Group the first two terms, and also group the second two terms. Take out any common factor from the group.
x2(x – 1) + 7(x – 1)
Notice x – 1 is a common factor.
(x – 1)(x2 + 7)
It should be pointed out that two groupings will work, but the third does not. So, if pairing terms one way does not work it may be that pairing terms in a different way will be worth doing. In an advanced approach terms can be grouped three or more at a time into a single group.
Factoring a trinomial is not always possible, but the technique to factor trinomials is well worth learning.
Factor x2 - 3x – 4
Note that the negative signs are parts of the numbers. Algebra is built on two operations, addition and multiplication, and subtraction is thought of as adding a negative number. Likewise, division is thought of as multiplying by a reciprocal.
We want two numbers that add to -3, and multiply as -4.
(x + a)(x + b) = x2 + ax + bx + ab = x2 + (a + b)x + ab
So, we need two numbers that multiply together to give, in this case, -4 and 1. The possible number pairs are (-1, 4), (-2, 2), and (-4, 1). In order to get the middle term to be -3x, we need that pair of numbers from the list that add to -3. That is -4 and 1.
x2 - 3x – 4 factors as (x – 4)(x + 1).
When x2 has a coefficient it also must also be factored. However, changing all of the signs gives nothing new, so it need not be done for the lead term.
Factor 2x2 – 5x + 3.
The possible pair for the 2 is (2, 1). (Notice -2, -1 will lead to repetitions.) The possible pairs for the 3 are (-1, -3) and (1, 3). Now we must be careful where we place things. The -5 must come from multiplying factors of the 2 by factors of the 3. 2(-1) and 1(-3) add together to get
-2 + (-3) = -5.
Remember, we are undoing polynomial multiplication.
(2x - 3)(x - 1) has the 2(-1) and the 1(-3). Proper placement is critical.