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Cubic trinomials are more difficult to factor than quadratic polynomials, mainly because there is no simple formula to use as a last resort as there is with the quadratic formula. (There is a cubic formula, but it is absurdly complicated). For most cubic trinomials, you will need a graphing calculator.
Cubic Trinomials of the Form Ax^3 + Bx+^2 + Cx
Step 1
Extract the greatest common factor of the trinomial. This is equal to k times x, where k is the greatest common factor of the three constant coefficients A, B and C of the polynomial. For example, the greatest common factor of the trinomial 3x^3 – 6x^2 – 9x is 3x, so the polynomial is equal to 3x times the trinomial x^2 – 2x -3, or 3x*(x^2 – 2x – 3).
Step 2
Factor the quadratic polynomial Ax^2 + Bx + C in the above polynomial by finding two numbers whose sum is equal to B and whose product is equal to A times C. For example, the polynomial x^2 – 2x – 3 factors as (x – 3)(x + 1).
Step 3
Write the factored form of the cubic trinomial by multiplying the GCF (found in Step 1) by the factored form of the polynomial. For example, the above polynomial is equal to 3x*(x – 3)(x – 1).
Other Cubic Trinomials
Step 1
Graph the polynomial on your calculator. Guess the values of the x-intercepts (points where the graph of the line crosses the x-axis). Check your guess by substituting these values of x into the trinomial one at a time. If the trinomial equals zero, the x value is an intercept.
Step 2
Verify that the x-intercepts are correct by dividing the polynomial by the binomial (x – a), where a is equal to the x value of the x-intercept you are testing. A simple way to divide polynomials is synthetic division. The binomial (x – a) is a factor of the polynomial if and only if it divides with a remainder of zero.
Step 3
Once you have verified that all the x-intercepts are correct, rewrite the polynomial in factored form as (x – a)(x – b)(x – c), where a, b and c are the x-intercepts of the equation. Some of the intercepts may be repeated, in which case the factored form will be (x – a)(x-b)^2 or (x – a)^3.
