Mathematics.

expressions

Factoring

Algebra I30 minDifficulty3 out of 10

You should know: polynomials

Overview

Factoring is the process of rewriting a polynomial as a product of simpler polynomials — the reverse of expanding/multiplying. It is one of the most useful skills in algebra because it converts sums (which are hard to solve or simplify) into products (whose zeros and common factors are easy to read off). Factoring is essential for solving polynomial equations, simplifying rational expressions, and revealing the structure hidden inside an expression.

Intuition

Factoring is like reverse-engineering a multiplication problem: if you know 12 = 3 × 4, factoring 12 means finding those numbers 3 and 4 starting only from 12. For polynomials it's the same idea in reverse of FOIL — given x² + 5x + 6, factoring asks 'what two binomials multiply together to give this?' The payoff is the zero-product property: if a product equals zero, at least one factor must be zero, which turns a hard equation into two easy ones.

Interactive Graph

x^2-5x+6 — the roots are exactly the factors' zeros

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Formal Definition

Definition

Factoring a polynomial means expressing it as a product of polynomials of lower degree (ideally irreducible ones). The most common patterns used in Algebra 1:

ab+ac=a(b+c)ab+ac = a(b+c)
GCF factoring
a2−b2=(a−b)(a+b)a^2-b^2=(a-b)(a+b)
Difference of squares
x2+bx+c=(x+p)(x+q) where p+q=b, pq=cx^2+bx+c=(x+p)(x+q) \text{ where } p+q=b,\ pq=c
Trinomial factoring

Derivation

The trinomial-factoring pattern comes directly from reversing FOIL: expanding (x+p)(x+q) shows exactly which sum/product conditions p and q must satisfy.

(x+p)(x+q)=x2+qx+px+pq=x2+(p+q)x+pq(x+p)(x+q) = x^2+qx+px+pq = x^2+(p+q)x+pq

Expand a general binomial product

So factoring x2+bx+c means finding p,q with p+q=b and pq=c\text{So factoring } x^2+bx+c \text{ means finding } p,q \text{ with } p+q=b \text{ and } pq=c

Match coefficients to reverse the expansion

Properties

Zero-product property

ab=0  ⟺  a=0 or b=0ab=0 \iff a=0 \text{ or } b=0

Difference of squares

a2−b2=(a−b)(a+b)a^2-b^2=(a-b)(a+b)

Perfect square trinomial

a2±2ab+b2=(a±b)2a^2\pm 2ab+b^2=(a\pm b)^2

Sum/difference of cubes

a3±b3=(a±b)(a2∓ab+b2)a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2)

Applications

Factoring characteristic polynomials of linear systems reveals natural frequencies and stability behavior directly from the roots.

Worked Examples

  1. Find two numbers that multiply to 12 and add to 7: 3 and 4.

    x2+7x+12=(x+3)(x+4)x^2+7x+12=(x+3)(x+4)

Answer: (x + 3)(x + 4)

Practice Problems

Difficulty 3/10

Factor: x² - 2x - 15.

Difficulty 4/10

Factor completely: 3x² + 12x + 12.

Common Mistakes

Common Mistake

Forgetting to factor out the greatest common factor (GCF) first, e.g. trying to factor 2x²+8x+6 directly as a trinomial without pulling out the 2.

Always check for a GCF across ALL terms first. 2x²+8x+6 = 2(x²+4x+3) = 2(x+1)(x+3) — skipping the GCF step often makes trinomial factoring impossible or leads to an incomplete factorization.

Common Mistake

Believing a² + b² factors like a² - b² does, e.g. writing x²+4 as (x+2)(x-2).

The sum of two squares a²+b² does NOT factor over the real numbers (unlike the difference a²-b², which factors as (a-b)(a+b)). Check by expanding: (x+2)(x-2)=x²-4, not x²+4.

Summary

  • Factoring rewrites a polynomial as a product of simpler polynomials, reversing expansion.
  • Always check for a greatest common factor (GCF) first.
  • Key patterns: difference of squares a²-b²=(a-b)(a+b), perfect square trinomials, and sum/difference of cubes.
  • The zero-product property (ab=0 ⟹ a=0 or b=0) is why factoring is used to solve polynomial equations.

References

  1. WebsiteWikipedia — Factorization