Before looking at some further factoring expressions and factoring binomials examples, it helps to be clear about what exactly a factor is in Math.
Firstly, we can recap what a factor is when we are dealing with numbers.
Factors with Numbers:
A factor of a certain number, is any other number that exactly divides the certain number, leaving no remainder.This includes the certain number itself.
We could have the number 16.
The factors of 16 are: 1 , 2 , 4 , 8 and 16.
\bf{\frac{16}{1}} \space = \space 16 , \bf{\frac{16}{2}} \space = \space 8 , \bf{\frac{16}{4}} \space = \space 4 , \bf{\frac{16}{8}} \space = \space 2 , \bf{\frac{16}{16}} \space = \space 1
So we can multiply some factors together to get the original number, such as 2 × 8 = 16.
This same logic and process also applies to expressions as well as numbers.
Factors with Expressions:
Similar to numbers, when dealing with factoring expressions examples we are looking to separate an expression into factors that can be multiplied together.With the expression 2x+6, we can factor.
What we want to do, is look for a common factor that both terms share, if we recall the distributive property,
a(b+c) \space = \space ab+ac.
2x+6 has a common factor of 2, so we can factor as follows.
2x + 6 \space = \space 2(x + 3)
The factors of 2x+6 are 2 and (x+3).
Factoring Binomials Examples
A binomial is an expression in Math that contains two terms.
(1.1)
Factor 2a^2 + 8a^3.
Solution
2a^2 is a common factor of both terms.
2a^2 + 8a^3 \space = \space 2a^2(1 + 4a)
(1.2)
Factor x^4 - 36.
Solution
We can look to put both terms into square form.
x^4 - 36 \space = \space (x^2)^2 - 6^2
Now from here we can recall the difference of two squares property.
(x^2)^2 - 6^2 \space = \space (x^2 + 6)(x^2 - 6)
(1.3)
Factor 25c^2d^4 - 4e^2.
Solution
25c^2d^4 - 4e^2 \space = \space (5cd^2)^2 - (2e)^2
= \space (5cd^2 - 2e)(5cd^2 + 2e)