Factor of a Polynomial

High School

Definition

A polynomial divides into other polynomials. The process of factoring a polynomials breaks the polynomial into products of smaller polynomials. By multiplying these products together, you'd be able to return to the original polynomial.

Worked examples

\(x^2 + 5x + 6 = (x + 2)(x + 3)\)
The binomials \((x + 2)\) and \((x + 3)\) are factors; multiplying them returns the original polynomial.
\(2x^3 - 8x = 2x(x^2 - 4) = 2x(x - 2)(x + 2)\)
Factor out the GCF first, then continue factoring until each factor is irreducible.
\(x^2 - 9 = (x - 3)(x + 3)\)
Difference of squares factors into two binomials with opposite middle signs.

Common mistakes

  • \(x^2 + 9 = (x + 3)(x + 3)\) → \(x^2 + 9\) does not factor over the reals Sum of squares does not factor; only difference of squares factors.
  • \(x^2 + 5x + 6 = (x + 1)(x + 6)\) → \(x^2 + 5x + 6 = (x + 2)(x + 3)\) Check by multiplying back: the pair must multiply to 6 and add to 5.
  • Stop at \(2x(x^2 - 4)\) when factoring \(2x^3 - 8x\) → Factor completely: \(2x(x - 2)(x + 2)\) Always factor until no polynomial factor can be broken down further.

Where you'll use it next

Factoring polynomials is essential for solving polynomial equations, simplifying rational expressions, graphing by finding intercepts, and tackling calculus topics like integration by partial fractions.

Found in 4 StudyPug lessons

Factoring Trinomials: Mastering x^2 + bx + c

Algebra 1

Unlock the secrets of factoring trinomials with our comprehensive guide. Learn powerful techniques like decomposition and cross-multiplication to solve x^2 + bx + c effortlessly.

10th Grade10th
Mastering Common Factors of Polynomials

Algebra 1

Unlock the power of polynomial factoring! Learn to identify common factors, simplify complex expressions, and solve equations with confidence. Boost your algebra prowess with our step-by-step guide and practice problems.

10th Grade10th
Factoring Polynomials: Mastering ax^2 + bx + c

Grade 10 Math

Unlock the secrets of factoring complex polynomials with our expert guide. Learn multiple methods, tackle challenging problems, and boost your algebra skills today!

10th Grade10th
Factorising polynomials by grouping

Algebra 1

There are a number of ways to factor polynomials, and one of them is by grouping. When using this grouping method, we will need to look for any common factors and then rewrite them as grouped factors.

10th Grade10th

See also

Degree of a PolynomialFactor TheoremBinomialQuadraticFactoring RulesZero of a Function

Reviewed by Pat Cheng, M.Ed. — StudyPug Curriculum Lead · Last updated June 6, 2026

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