Factoring Calculator

Factoring (or factorisation) is the process of breaking a polynomial expression into a product of simpler expressions called factors. It is the reverse of expanding (multiplying out).

For example:

$x^2 - 5x + 6 = (x - 2)(x - 3)$

The left side is a single polynomial; the right side is the same expression written as a product of two binomials.

Factoring is essential in algebra because it allows us to:

• Solve equations: Setting each factor to zero gives the roots.
• Simplify fractions: Cancel common factors in rational expressions.
• Analyze behavior: Identify zeros, asymptotes, and sign changes.

A polynomial is fully factored when each factor is irreducible (cannot be factored further over the integers). The Fundamental Theorem of Algebra guarantees that every polynomial of degree $n$ can be factored into exactly $n$ linear factors over the complex numbers.

Common types of factoring include:

• Factoring out the Greatest Common Factor (GCF)
• Factoring trinomials
• Difference of squares: $a^2 - b^2 = (a+b)(a-b)$
• Sum/difference of cubes
• Factoring by grouping

Here are the main factoring techniques, ordered from simplest to most advanced:

1. Factor Out the GCF

Always start by pulling out the greatest common factor.

Example: $6x^3 + 9x^2 = 3x^2(2x + 3)$

2. Difference of Squares

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

Example: $x^2 - 16 = (x + 4)(x - 4)$

3. Perfect Square Trinomials

$a^2 + 2ab + b^2 = (a + b)^2$
$a^2 - 2ab + b^2 = (a - b)^2$

Example: $x^2 + 6x + 9 = (x + 3)^2$

4. Trinomial Factoring ($x^2 + bx + c$)

Find two numbers $p$ and $q$ such that $p + q = b$ and $p \cdot q = c$:

$x^2 + bx + c = (x + p)(x + q)$

Example: $x^2 - 5x + 6$: find $p + q = -5$ and $pq = 6$ → $p = -2, q = -3$

So $x^2 - 5x + 6 = (x - 2)(x - 3)$

5. AC Method (for $ax^2 + bx + c$ with $a \neq 1$)

Multiply $a \cdot c$, find two numbers that multiply to $ac$ and add to $b$, then split and group.

Example: $2x^2 + 7x + 3$: $ac = 6$, find $1 + 6 = 7$

• $2x^2 + x + 6x + 3 = x(2x+1) + 3(2x+1) = (x+3)(2x+1)$

6. Sum/Difference of Cubes

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

7. Factoring by Grouping

Group terms in pairs and factor each pair, then factor out the common binomial.

• Forgetting to factor out the GCF first: Always check for a common factor before using other techniques.
• Confusing difference vs. sum of squares: $a^2 - b^2$ factors, but $a^2 + b^2$ does not factor over the reals.
• Sign errors in trinomial factoring: When $c > 0$ and $b < 0$, both $p$ and $q$ are negative.
• Stopping too early: Check if each factor can be factored further (e.g., $x^4 - 16 = (x^2+4)(x^2-4) = (x^2+4)(x+2)(x-2)$).
• Not verifying by expanding: Always multiply your factors back out to confirm they equal the original expression.