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 \( x^4 - y^4 = (x^2 - y^2)(x^2 + y^2) = (x-y)(x+y)(x^2+y^2) \)