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How to Factor Cubic Polynomials Using Long Division?

Published in Polynomial Factoring 3 mins read

Factoring cubic polynomials using long division is a method used when you know at least one root (or factor) of the polynomial. Here's a breakdown of the process:

Steps to Factor a Cubic Polynomial Using Long Division

  1. Find One Root (or Factor): This is the crucial first step. You need to know one value of x that makes the polynomial equal to zero. This can be found through:

    • Trial and Error: Substitute simple values like 0, 1, -1, 2, -2, etc., into the polynomial. If the result is zero, you've found a root.
    • Rational Root Theorem: This theorem can help you identify potential rational roots. It states that if a polynomial has integer coefficients, then any rational root will be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
    • Given Information: Sometimes, the problem will provide you with a root.

  2. Determine the Corresponding Factor: Once you have a root, say x = a, you know that (x - a) is a factor of the polynomial.

  3. Set up the Long Division: Write the cubic polynomial as the dividend and the factor (x - a) as the divisor.

  4. Perform Long Division:

    • Divide the first term of the dividend by the first term of the divisor. This gives you the first term of the quotient.
    • Multiply the entire divisor by the first term of the quotient.
    • Subtract the result from the dividend.
    • Bring down the next term of the dividend.
    • Repeat the process until you have no more terms to bring down.

  5. Interpret the Result:

    • If the division is done correctly, the remainder should be zero.
    • The quotient will be a quadratic polynomial.

  6. Factor the Quadratic: Factor the resulting quadratic polynomial (quotient) using any standard factoring method (factoring by grouping, quadratic formula, etc.).

  7. Write the Factored Form: The cubic polynomial is now factored as the product of the linear factor (x - a) and the factored quadratic polynomial.

Example

Let's say you have the cubic polynomial x³ - 6x² + 11x - 6 and you know that x = 1 is a root.

  1. Root: x = 1

  2. Factor: (x - 1)

  3. Long Division:

            x²  - 5x + 6
  x - 1 | x³ - 6x² + 11x - 6
        -(x³ - x²)
        -----------------
             -5x² + 11x
             -(-5x² + 5x)
             -----------------
                      6x - 6
                      -(6x - 6)
                      -----------------
                              0

  4. Quotient: The quotient is x² - 5x + 6.

  5. Factor the Quadratic: x² - 5x + 6 = (x - 2)(x - 3)

  6. Factored Form: Therefore, x³ - 6x² + 11x - 6 = (x - 1)(x - 2)(x - 3)

Summary

Factoring cubic polynomials using long division involves finding a root (and its corresponding factor), dividing the cubic polynomial by that factor, and then factoring the resulting quadratic. This breaks down the cubic into simpler, more manageable factors. The availability of one root is key to efficiently employing this method.

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