How to Factor Cubed Polynomials

Use several methods to factor cubic polynomials.

Factoring polynomials has been of great interest to mathematicians for centuries. Finding factors of a polynomial is equivalent to finding the roots of the associated equation, which is a key aim in algebra. A number of methods have been devised to find roots for polynomials of various orders, including cubic polynomials.

Instructions

Grouping

1. Express the cubic polynomial in the standard form of: ax^3 + bx^2 + cx +d, where "^" implies "raised to the power of." Note that "b," "c" or "d" may be zero but "a" cannot be. Otherwise, the polynomial is no longer a cubic.

2. Separate the terms into two groups of the form (ax^3 + bx^2) + (cx +d).

3. Extract, in turn, the greatest common factor, or GCF, of each of the first group: (ax^3 + bx^2) and the second: (cx + d) separately, where they exist, and express each group in factored form. Note that x^2 will be part of any factor of the first group of terms.

4. Extract the GCF, where it exists, of the first and second groups combined. The ideal result will be in the form: (x - g)(x - h)(x - i), although this may not be achievable in all cases. Multiply out the terms to verify the correctness of the factoring.

Reduction to Quadratic

1. Look for an obvious factor of the polynomial. The Factor Theorem states that if a polynomial f(x) has a root g, such that f(g) = 0, then that polynomial has a factor (x - g).

   Try, in turn, values such as 0, +1, -1, +2, -2. Where a value, say x = g, is found that reduces the polynomial to zero, divide the original polynomial by (x - g) and factor the results in the form (x - g)(px^2 + qx + r). Note that the polynomial in the second bracket is now a quadratic.

2. Repeat Step 1 to see if there is another obvious factor for the quadratic polynomial, and factor this out to give the ideal form of (x - g)(x - h)(x - i).

3. Use the quadratic formula of (-q + or -- √(q^2 - 4pr))/2p, for the quadratic polynomial in Step 1, if no further factors are found in Step 2. This will give the other two factors: (-q + √(q^2 - 4 pr))/2p and (-q - √(q^2 - 4pr))/2p.

Tips & Warnings

• Note that every cubic equation has three roots and therefore three factors. But the roots may be whole numbers, fractions, irrational numbers or complex numbers.

• It may not always be possible to get the roots of a polynomial by analytical methods. In this case using graphical or numerical methods may give sufficiently accurate results.

References

• West Texas A&M University: Tutorial 7: Factoring Polynomials
• Think Quest: Algebra II: Polynomials and Factoring
• Oakroadsystems.com: Solving Polynomial Equations

Resources

• West Texas A&M University: Virtual Math Lab
• S.O.S. Math: The "Cubic formula"