# Steps to Factoring

These guidelines are for an Intermediate Algebra Level of understanding.

## Greatest Common Factor

Always begin by factoring out the greatest common factor (GCF) if it is anything other than 1.

$$15x^3-35x^2-30x \Longrightarrow 5x ( 3x^3-7x-6 )$$

## Look at the number of terms

### If there are two terms

• Difference of two squares, $$a^2-b^2 = (a-b)(a+b)$$
• Sum of two squares, $$a^2 + b^2$$ is prime. There are some exceptions, but they involve exponents greater than 2 or complex numbers.
• Difference of two cubes, $$a^3 - b^3 = (a-b)(a^2 + ab + b^2)$$
• Sum of two cubes, $$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$$

The difference of two squares and difference of two cubes are special cases of the broader difference of two nth-degree terms.

• $$a^4 - b^4 = (a-b)(a^3 + a^2b + ab^2 + b^3)$$
• $$a^5 - b^5 = (a-b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)$$
• $$a^n - b^n = (a-b)(a^{n-1} + a^{n-2}b + a^{n-3}b^2 \dots + a^2b^{b-3} + ab^{n-2} + b^{n-1} )$$

### If there are three terms

• Perfect square trinomial factors as a binomial squared, $$a^2 + 2ab + b^2 = (a+b)^2$$
• Trial and error method or the ac method of factoring, $$ax^2 + bx + c$$

### If there are four or more terms

• Grouping (might involve rearranging the order of terms as well)
• Grouping in pairs, $$3x^3 + 5x^2 - 6x - 10 = x^2(3x+5) - 2(3x+5) = (x^2-2)(3x+5)$$
• Look for perfect square trinomials, $$4x^2 + 12x + 9-4y^2 = (4x^2 + 12x + 9)-4y^2 = (2x+3)^2-(2y)^2 = (2x+3-2y)(2x+3+2y)$$

## Check for futher factorability

After you have factored, check what remains to see if it can be factored more.

For example, if you choose to factor $$x^4 - y^4$$ as the difference of two squares, you get: $$x^4 - y^4 = (x^2 - y^2)(x^2 + y^2)$$, but the $$x^2 - y^2$$ can be factored further as $$(x-y)(x+y)$$. So we get $$x^4 - y^4 = (x^2 - y^2)(x^2 + y^2) = (x-y)(x+y)(x^2+y^2)$$