Solve Polynomial Equations by Factoring
Solving polynomial equations by factoring is a Grade 11 algebra skill in Big Ideas Math. After setting the polynomial equal to zero, the goal is to factor it completely and then apply the Zero Product Property: if A × B = 0, then A = 0 or B = 0. For example, x³ − 4x = 0 factors as x(x − 2)(x + 2) = 0, giving solutions x = 0, 2, −2. Common factoring strategies include GCF extraction, difference of squares, sum/difference of cubes, and trinomial factoring. For degree ≥ 3, grouping or the Rational Root Theorem helps identify factors before applying the Zero Product Property.
Key Concepts
Polynomial Equation.
An equation of the form $P(x) = 0$ where $P(x)$ is a polynomial function is called a polynomial equation.
Common Questions
What is the Zero Product Property?
If the product of two or more factors equals zero, then at least one of the factors must equal zero. For A × B = 0: A = 0 or B = 0 (or both).
How do you solve x³ − 4x = 0 by factoring?
Factor out GCF: x(x² − 4) = 0. Factor the difference of squares: x(x − 2)(x + 2) = 0. Apply Zero Product Property: x = 0, x = 2, x = −2.
What is the first step in solving any polynomial equation by factoring?
Move all terms to one side so the equation equals zero, then factor completely using GCF, patterns (difference of squares, trinomial), or grouping.
What is the difference of squares pattern?
a² − b² = (a + b)(a − b). For example, x² − 9 = (x + 3)(x − 3).
How do you factor a cubic polynomial?
Try GCF first, then test rational roots using the Rational Root Theorem to find one factor, then divide the cubic by (x − root) to get a quadratic that can be factored further.
Can every polynomial be solved by factoring?
No—polynomials with irrational or complex roots cannot be factored over integers. In those cases, use the quadratic formula, completing the square, or numerical methods.