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Mom's Basement University

Factoring Higher-Degree Polynomials

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn how to factor polynomials of degree 3 and higher using grouping and the rational root theorem.

The Concept

Section titled “The Concept”

Factoring higher-degree polynomials builds on what you learned with quadratics. The goal is to write the polynomial as a product of lower-degree factors.

Common strategies:

  1. Greatest Common Factor (GCF) - always check first.
  2. Grouping - useful for polynomials with 4 or more terms.
  3. Rational Root Theorem - helps identify possible rational roots to test.

Rational Root Theorem: any possible rational root p/q has p as a factor of the constant term and q as a factor of the leading coefficient.

Once you find a root, use synthetic division to factor it out and reduce the degree.

Worked Example

Section titled “Worked Example”

Factor 2x³ − 3x² − 11x + 6.

The graph shows the cubic crossing the x-axis at three points. These are the roots we need to find. Each gold dot is a solution to 2x³ − 3x² − 11x + 6 = 0. The factored form (x − 3)(2x − 1)(x + 2) tells us exactly where those crossings happen.

1. List possible rational roots

The constant term is 6 and the leading coefficient is 2. By the Rational Root Theorem, possible rational roots are ±1, ±2, ±3, ±6, ±1/2, ±3/2.

2. Test x = 3

2(3)3−3(3)2−11(3)+6=54−27−33+6=0  ✓2(3)^3 - 3(3)^2 - 11(3) + 6 = 54 - 27 - 33 + 6 = 0 \;\checkmark

So x = 3 is a root, which means (x − 3) is a factor.

3. Use synthetic division to divide by (x − 3)

Synthetic division is a shortcut for dividing a polynomial by a linear factor (x − r). Instead of doing long division, you only work with the coefficients. Here’s how it works step by step for dividing 2x³ − 3x² − 11x + 6 by (x − 3):

  1. Write the root (3) on the left and the coefficients of the polynomial (2, −3, −11, 6) in a row.
  2. Bring down the first coefficient (2).
  3. Multiply the root by that number: 3 × 2 = 6. Write it under the next coefficient.
  4. Add down the column: −3 + 6 = 3.
  5. Repeat: 3 × 3 = 9, write under next coefficient, add: −11 + 9 = −2.
  6. Repeat once more: 3 × (−2) = −6, add: 6 + (−6) = 0. The zero confirms x = 3 is a root.
32−3−116
69−6
23−20

The bottom row (2, 3, −2) gives the coefficients of the quotient: 2x² + 3x − 2. The last number (0) is the remainder. Zero means (x − 3) divides evenly.

4. Factor the remaining quadratic

2x2+3x−2=(2x−1)(x+2)2x^2 + 3x - 2 = (2x - 1)(x + 2)

Final factored form:

2x3−3x2−11x+6=(x−3)(2x−1)(x+2)2x^3 - 3x^2 - 11x + 6 = (x - 3)(2x - 1)(x + 2)

The roots are x = 3, x = 1/2, and x = −2, matching the three gold dots on the graph.

Real-World Application

Section titled “Real-World Application”

Factoring higher-degree polynomials helps in:

  • Finding break-even points in business models
  • Solving volume or area optimization problems
  • Analyzing projectile or motion equations
  • Engineering design (beam stress, fluid flow)

Example: a company’s profit model is a cubic polynomial. Factoring it helps find the production levels where profit is zero (break-even points).

Quiz

Section titled “Quiz”
According to the Rational Root Theorem, possible rational roots are factors of the constant term over factors of the:
What is a good first step when factoring a cubic polynomial?
After finding one root of a cubic, the next step is usually:
The polynomial $x^3 - 6x^2 + 11x - 6$ factors as:
Factor $x^3 + 2x^2 - 5x - 6$ given that $x = -1$ is a root.