Mathematics ยท Algebra

Factoring Techniques for Polynomials

Reference entry · last updated September 20, 2026

Factoring a polynomial means writing it as a product of polynomials of lower degree. It reverses multiplication and is the main tool for solving polynomial equations, because a product equals zero only when at least one factor equals zero.

1. First principles

Factoring relies on the distributive law read in reverse: \( ab + ac = a(b + c) \). The zero-product property then states that if a product of real factors is zero, at least one factor is zero. Together they convert a hard root-finding problem into simpler ones [1, 2].

2. Greatest common factor

Always try the greatest common factor first. Factor out the largest monomial dividing every term. Example: \( 6x^3 - 9x^2 = 3x^2(2x - 3) \).

3. Difference of squares

\( a^2 - b^2 = (a - b)(a + b) \)

A sum of squares, \(a^2 + b^2\), does not factor over the real numbers. The difference-of-squares pattern can be applied more than once, as in \( x^4 - 16 = (x^2 - 4)(x^2 + 4) = (x - 2)(x + 2)(x^2 + 4) \).

4. Perfect square trinomials

\( a^2 + 2ab + b^2 = (a + b)^2, \qquad a^2 - 2ab + b^2 = (a - b)^2 \)

Recognize the pattern by checking that the first and last terms are perfect squares, written \(A^2\) and \(B^2\), and that the middle term equals \(2AB\) or \(-2AB\). Using the square roots rather than the terms themselves is what makes the rule correct: in \(x^2 + 6x + 9\) the square roots are \(x\) and \(3\), and \(2 \cdot x \cdot 3 = 6x\).

5. Sum and difference of cubes

\( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)
\( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)

The quadratic factor does not factor further over the reals. Keep the signs straight: the linear factor takes the sign of the cube, and the middle term of the quadratic takes the opposite sign.

6. General trinomials and grouping

For \( x^2 + bx + c \), find two numbers whose product is \(c\) and whose sum is \(b\), then write \( (x + p)(x + q) \).

For \( ax^2 + bx + c \) with \( a \neq 1 \), use the \(ac\) method: find two numbers whose product is \(ac\) and whose sum is \(b\), split the middle term, and factor by grouping. Example: \( 2x^2 + 7x + 3 \) becomes \( 2x^2 + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) \).

With four terms, group in pairs and factor each pair: \( x^3 + 3x^2 + 2x + 6 = x^2(x + 3) + 2(x + 3) = (x^2 + 2)(x + 3) \).

7. Strategy

  1. Factor out the GCF.
  2. Count the terms. Two terms suggest difference of squares or cubes; three terms suggest a trinomial pattern; four terms suggest grouping.
  3. Factor again until every factor is irreducible.
  4. Check by multiplying the factors back and comparing with the original.

See also

References

  1. ^OpenStax, College Algebra 2e, Rice University (Factoring polynomials).
  2. ^Michael Sullivan, Algebra and Trigonometry, 11th ed., Pearson, 2019.