Factoring polynomials means rewriting a polynomial as a product of simpler expressions (its factors). This is the reverse of expanding: instead of multiplying to get a sum, the goal is to express the sum as multiplication.
Definition A polynomial \(P(x)\) is factored when it is written as \(P(x)=A(x)\cdot B(x)\cdot \dots\), where each factor is simpler than the original polynomial and multiplying them reproduces \(P(x)\).
A systematic checklist for factoring polynomials
A reliable approach to factoring polynomials uses a sequence of tests. Skipping early tests often leads to harder work later.
| Step | What to check | Typical outcome |
|---|---|---|
| 1 | Greatest common factor (GCF) of all terms | Factor out a monomial, leaving a simpler polynomial |
| 2 | Special patterns (difference of squares, perfect square trinomial) | Recognize and apply a known identity |
| 3 | Trinomial factoring \(ax^2+bx+c\) | Write as \((px+q)(rx+s)\) with correct products and sums |
| 4 | Factoring by grouping (4 terms or structured expression) | Create a common binomial factor |
| 5 | Check for further factoring (repeat steps on remaining factors) | Fully factored form over integers/rationals |
Key identities used in factoring polynomials
\[ a^2-b^2=(a-b)(a+b) \]
\[ a^2+2ab+b^2=(a+b)^2,\qquad a^2-2ab+b^2=(a-b)^2 \]
Worked examples of factoring polynomials
Example 1: Factor out the greatest common factor (GCF)
Factor \(6x^3-9x^2\).
The GCF of \(6x^3\) and \(-9x^2\) is \(3x^2\), because \(\gcd(6,9)=3\) and the smallest power of \(x\) is \(x^2\).
\[ 6x^3-9x^2 = 3x^2(2x-3) \]
Example 2: Difference of squares
Factor \(x^2-16\).
Recognize \(x^2-16=x^2-4^2\), which matches the difference of squares identity.
\[ x^2-16=(x-4)(x+4) \]
Example 3: Trinomial factoring
Factor \(x^2+5x+6\).
Look for two numbers whose product is \(6\) and whose sum is \(5\). The pair \(2\) and \(3\) works.
\[ x^2+5x+6=(x+2)(x+3) \]
Example 4: Factoring by grouping
Factor \(x^3+3x^2+2x+6\).
Group terms: \((x^3+3x^2)+(2x+6)\). Factor each group:
\[ x^3+3x^2 = x^2(x+3),\qquad 2x+6=2(x+3) \]
A common binomial factor \((x+3)\) appears:
\[ x^3+3x^2+2x+6 = x^2(x+3)+2(x+3) = (x+3)(x^2+2) \]
Visualization: factoring as “area splitting”
A common way to interpret factoring polynomials is to view a product as the area of a rectangle. Expanding corresponds to splitting the rectangle into smaller rectangles; factoring reverses that process by identifying the side lengths.
How to confirm a factorization is correct
The standard verification step in factoring polynomials is to multiply the proposed factors and simplify. If the result matches the original polynomial exactly, the factorization is correct.
\[ (x+2)(x+3)=x^2+3x+2x+6=x^2+5x+6 \]