📖Definition
Factoring means rewriting an expression as a product of simpler expressions, called factors, that multiply together to give the original. It is the reverse of expanding brackets, and it is a key skill for solving quadratic equations.
📐Formula
This is the difference of squares pattern: any expression that is one perfect square minus another perfect square factors into a sum times a difference.
📝Step-by-Step Guide
Check for a greatest common factor (GCF)
Look for a number or variable that divides every term, and pull it outside a bracket first.
Factor trinomials by finding two numbers
For x² + bx + c, find two numbers that multiply to c and add to b.
Recognize special patterns
Watch for the difference of squares and perfect square trinomials, since they factor in a predictable way.
Check by expanding
Multiply your factors back out to confirm they reproduce the original expression.
🧭Factoring is the reverse of expanding
Expanding (x + 2)(x + 3) gives x² + 5x + 6. Factoring reverses that process, starting from x² + 5x + 6 and finding the two brackets that produced it.
This is why checking your answer by expanding it back out is such a reliable way to catch mistakes.
🧭Why factoring matters for solving equations
If a product of two factors equals zero, at least one of the factors must be zero. This is called the zero-product property.
That is exactly how factoring is used to solve quadratic equations: factor the expression, set each bracket equal to zero, and solve each simple equation.
⚠️Common Mistakes to Avoid
- Forgetting to check for a GCF before trying other methods
- Sign errors when the constant term is negative
- Stopping before the expression is fully factored
- Trying to apply the difference of squares to a sum of squares, which does not factor over the real numbers
- Losing track of which two numbers were needed and guessing instead
✏️Practice Problems
Factor x² + 5x + 6
💡 Hint: Find two numbers that multiply to 6 and add to 5.
Answer: (x + 2)(x + 3)
Factor 2x² + 8x
💡 Hint: Look for a common factor first.
Answer: 2x(x + 4)
Factor x² − 9
💡 Hint: This is a difference of squares.
Answer: (x − 3)(x + 3)
Factor x² + 6x + 9
💡 Hint: Both numbers you need are the same.
Answer: (x + 3)²
Factor x² − 2x − 15
💡 Hint: You need two numbers that multiply to −15 and add to −2.
Answer: (x − 5)(x + 3)
Factor 3x² + 12x + 9 completely
💡 Hint: Take out the common factor of 3 first.
Answer: 3(x + 1)(x + 3)
Printable Factoring practice
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❓Factoring FAQ
What if the expression cannot be factored?
Not every quadratic factors neatly with whole numbers. When that happens, use the quadratic formula instead, which works on any quadratic equation.
How do I factor when the x² term has a coefficient greater than 1?
Look for a common factor first. If none exists, use a method such as splitting the middle term, where you rewrite bx as two terms that let you group and factor in pairs.
Is there a shortcut for spotting a difference of squares?
Yes. If you see one perfect square term minus another perfect square term, with no middle term, it factors as (first + second)(first − second).