# 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 n^{th}-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 \( x^4 - y^4 = (x^2 - y^2)(x^2 + y^2) = (x-y)(x+y)(x^2+y^2) \)