Factoring explained
Factoring rewrites a number or expression as a product of simpler pieces. It reverses multiplication and is a key step in solving equations and simplifying fractions. This guide covers common factors, quadratics, and the special patterns worth memorizing.
What factoring is
Just as 12 factors into 2 times 2 times 3, an algebraic expression like x squared plus 5x plus 6 factors into (x plus 2)(x plus 3). Factoring undoes multiplication, breaking an expression into the simpler pieces that were multiplied to make it.
Greatest common factor first
Always start by pulling out the greatest common factor, the largest term that divides every part. In 6x plus 9, both terms share a 3, so it factors to 3(2x plus 3). Taking out the common factor first often makes whatever remains much easier to factor further.
Factoring a quadratic
- Write the quadratic in the form x² + bx + c.
- Find two numbers that multiply to c and add to b.
- Write the factors as (x + first number)(x + second number).
- Check by multiplying the brackets back out.
Factor x² + 7x + 12. Two numbers multiplying to 12 and adding to 7 are 3 and 4, so it factors to (x + 3)(x + 4).
Special patterns
A few patterns are worth recognizing instantly. A difference of squares, a² minus b², factors to (a plus b)(a minus b). A perfect square trinomial like a² plus 2ab plus b² factors to (a plus b) squared. Spotting these saves time and shows up constantly in algebra.
- Factoring rewrites an expression as a product of simpler factors.
- Always pull out the greatest common factor first.
- To factor x² + bx + c, find two numbers that multiply to c and add to b.
- Difference of squares: a² − b² = (a + b)(a − b).
- Factoring is the reverse of multiplying out brackets.
Related guides
See How to Solve Quadratic Equations, where factoring is one of the methods.
FAQ
What does it mean to factor an expression?
To rewrite it as a product of simpler expressions, reversing multiplication. For example, x² + 5x + 6 factors to (x + 2)(x + 3).
How do I factor a quadratic?
Find two numbers that multiply to the constant term and add to the middle coefficient, then write the matching brackets.
What is a difference of squares?
An expression of the form a² − b², which always factors to (a + b)(a − b).