Introduction
Factoring is the reverse process to multiplication.
Exchanging money is another common function that relies on factoring like 4 quarters to make a dollar.
1. Common Factor
Each term of an expression contains the same quantity, which is called the common factor.
Example: Factor \(4y^3 + 12y \)
Common factors: \(2\cdot 2\cdot y=4y\)
By taking out common factors,
\(4y^3+12y = \color{red}{4y (y^2+3)}\)Check: \(\color{red}{4y(y^2+3)} = \color{black}4y\cdot y^2 + 4y\cdot3=4y^3+12y\)
2. Factoring Using Identities
Here are some common identities:
| Identities | Expanded Form | Factored Form |
|---|---|---|
| Difference of Two Squares | \(a^2-b^2\) | \((a+b)(a-b)\) |
| Perfect Square Trinomials | \(a^2\pm 2ab+b^2\) | \((a\pm b)^2\) |
| Sum of Two cubes | \(a^3+b^3\) | \((a+b)(a^2-ab+b^2)\) |
| Difference of Two cubes | \(a^3-b^3\) | \((a-b)(a^2+2ab+b2\) |
| Cube of a Binomial | \(a^3\pm 3a^2b+3ab^2\pm b^3\) | \((a\pm b)^3\) |
Example 1
\(25x^2-49y^2\)
\(
=(5x)^2\color{blue}-\color{black}(7y)^2 \hspace{60 pt} [a=5x\text{ and }b=7y; a^2-b^2]\\
=\color{red}{(5x+7y)(5x-7y)}\hspace{36 pt} \color{black}{[a=5x\text{ and }b=7y; (a+b)(a-b)]}
\)
3. Factoring Quadratic Trinomials (leading coefficient a = 1 case):
Example 1
\(25x^2-49y^2\)
\(
=(5x)^2\color{blue}-\color{black}(7y)^2 \hspace{60 pt} [a=5x\text{ and }b=7y; a^2-b^2]\\
=\color{red}{(5x+7y)(5x-7y)}\hspace{36 pt} \color{black}{[a=5x\text{ and }b=7y; (a+b)(a-b)]}
\)
