How to Factor Polynomials and the Difference of Cubes

Polynomials are functions that sum variables raised to different exponents, with constant coefficients--for example x^2+3x-4. The central aim of polynomial factoring is to rewrite them as the product of polynomials of lower order (i.e. lower variable exponent). For example, x^2+3x-4 = (x+4)(x-1). Several approaches are used to factor quadratics (polynomials with a maximum exponent of 2), such as grouping, reverse FOIL, and using the quadratic formula. Factoring the addition or difference of cubes is a matter of applying formulas, though there is a formula for cubics in general that involves two substitutions.

Instructions

Quadratic: By Grouping

1. Put the quadratic into the form ax^2+bx+c, where a, b, and c are constant coefficients.

2. Find factors of the product a---c that add to b.

   For example, for 3x^2+11x+10, a---c=30. Note that 6---5=30 and 6+5=b.

3. Separate b into the new addends you've found.

   For example, 3x^2+11x+10 = 3x^2+6x+5x+10.

4. Pull a common factor out of each of two pairs of terms.

   For example, 3x(x+2) + 5(x+2).

5. Factor out the common term.

   I.e. pull out x+2 to get (3x+5)(x+2).

Using Quadratic Formula

1. Put the quadratic into the form x^2+bx+c, i.e. a=1.

   To do this, the coefficient of x^2 can be pulled out of the whole polynomial and set to the side until it is multiplied back in at the end of the problem. For example, 3x^2+11x+10 can be written as x^2+11x/3+10/3.

2. Solve the quadratic formula for the two roots.

   The quadratic formula is x = [-b+-√(b^2-4ac)]/[2a]. For the example above, a=1, b=11/3, and c=10/3. So x = -2 and -5/3 are the roots of x^2+11x/3+10/3. I.e. x^2+11x/3+10/3 = (x-root1)(x-root2) = (x+2)(x+5/3).

3. Multiply back in the original "a."

   Our original a was 3. Multiplying that back in gives 3(x+2)(x+5/3) = (x+2)(3x+5), which is the same factorization we got by grouping. Note that using the quadratic formula may be slower, but has broader usefulness, when finding products of ac that add to give b is not easy, e.g. when the coefficients are fractions or large numbers.

Difference of Cubes

1. Put the difference in the form a^3-b^3.

2. Calculate a-b.

3. Calculate a^2+ab+b^2.

   Multiplying out the terms of (a-b)(a^2+ab+b^2) shows that it is equal to the original difference, a^3-b^3. An example is 27x^3-y^3. Let a=3x and b=y. Then the difference factors out to (3x-y)(9x^2+3xy+y^2).

Tips & Warnings

• Note that the formula for the addition of cubes is a^3+b^3=(a+b)(a^2-ab+b^2).

References

• Tidewater Community College: Types of Factoring