Factoring polynomials: a step-by-step guide

Factoring polynomials means writing a polynomial as a product of simpler polynomials — the reverse of expanding brackets. Expanding $(x + 2)(x + 3)$ gives $x^2 + 5x + 6$; factoring $x^2 + 5x + 6$ takes you back to $(x + 2)(x + 3)$. The product is called the factored form.

Factored form is useful: it shows where a polynomial equals zero (each factor gives a root), it simplifies fractions, and it makes graphs easier to sketch. In the UK the same skill is called factorising expressions.

Factoring steps: a checklist

1. Take out the greatest common factor (GCF) of all the terms.
2. Count the terms that are left.
3. Two terms: look for a difference of two squares, $a^2 - b^2 = (a + b)(a - b)$, or a sum or difference of two cubes.
4. Three terms (a trinomial): find two numbers that multiply to $a \times c$ and add to $b$.
5. Four terms: try grouping them in pairs.
6. Check each factor — can it be factored again? Then multiply out to check.

Common monomial factor: examples with answers

A common monomial factor is a single term that divides every term of the polynomial. Find the GCF of the numbers and the lowest power of each letter:

• $6x^2 + 9x = 3x(2x + 3)$
• $4x^3 - 8x^2 = 4x^2(x - 2)$
• $5a^2b + 10ab^2 = 5ab(a + 2b)$
• $7y - 21 = 7(y - 3)$

Factoring completely

Factoring completely means continuing until no factor can be broken down any further. The most common slip is stopping after the first step. In the example below, taking out $2x$ leaves $x^2 - 9$, which still factors.

Special patterns to recognise

• Perfect square trinomial: $a^2 + 2ab + b^2 = (a + b)^2$ and $a^2 - 2ab + b^2 = (a - b)^2$. Example: $x^2 - 10x + 25 = (x - 5)^2$.
• Difference of two squares: $a^2 - b^2 = (a + b)(a - b)$.
• Difference of two cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.
• Sum of two cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.

Why factored form is useful

Factored form turns hard equations into easy ones. To solve $x^2 + 5x + 6 = 0$, factor it as $(x + 3)(x + 2) = 0$. A product is zero only when a factor is zero, so $x = -3$ or $x = -2$. The same idea finds where a graph crosses the x-axis and simplifies algebraic fractions such as $\frac{x^2 - 9}{x + 3} = x - 3$ (for $x \neq -3$).

Common mistakes

• Skipping the common factor. Without it, $2x^3 - 18x$ looks much harder than it is.
• Sign errors in the two numbers: for $6x^2 + 7x - 3$ the product is $-18$, so one number must be negative.
• Stopping too early — always check whether each factor can be factored again.
• Treating a sum of squares like a difference: $x^2 + 25$ does not factor as $(x + 5)(x - 5)$, which expands to $x^2 - 25$.