Factoring
“Factoring is finding two numbers that 'multiply to the constant, add to the coefficient'.”
The formula
x² + (a + b)x + ab = (x + a)(x + b)How to read it: group using two numbers that add to the linear coefficient and multiply to the constant
- a + b
- — the coefficient of x = the sum of the two numbers
- ab
- — the constant term = the product of the two numbers
- (x + a)(x + b)
- — the factored product form
The hook
Factoring is reading the multiplication formulas backward — winding an expanded expression back into a 'product'.
In plain words
Turning something like x²+5x+6 into a 'product of two expressions' such as (x+2)(x+3). It's the exact reverse of expanding.
The intuition
It's recovering the 'width and height' of a rectangle whose area is x²+5x+6. Find the two numbers (2 and 3) that multiply to the constant 6 and add to the middle coefficient 5, and you're done.
How it's built
For the form x²+(a+b)x+ab, find two numbers a, b that 'add to the middle coefficient and multiply to the constant'. Then it factors as (x+a)(x+b). Match the signs too.
Example
x²+5x+6: two numbers multiplying to 6 and adding to 5 are 2 and 3 → (x+2)(x+3). Multiply back and it really is x²+5x+6.
Common mistake
Signs are easy to lose — x²−5x+6 multiplies to +6 and adds to −5, so it's −2 and −3, i.e. (x−2)(x−3). Also don't forget to pull out a common factor first (the 2x in 2x²+6x).
Where it's used
It's the fastest route to solving quadratic equations, and an all-purpose tool for cancelling fractions and simplifying messy expressions.
Where it came from
Splitting into 'factors' extends the ancient tradition of breaking numbers into prime factors, carried over to expressions with letters.
Prerequisites
Quick check
Factor x² − 9.
- (x + 3)(x − 3)✓
- (x − 3)²
- (x + 9)(x − 1)
- (x + 3)²
Practice
Factor x² + 7x + 12.
Answer: 0
- Two numbers multiplying to 12 and adding to 7: 3 and 4
- (x + 3)(x + 4)
Takeaway: Product = constant, sum = linear coefficient.
Factor x² − x − 6.
Answer: undefined
- Two numbers multiplying to −6 and adding to −1: −3 and 2
- (x − 3)(x + 2)
Takeaway: A negative constant → the two numbers have opposite signs.
Factor 2x² + 6x.
Answer: undefined
- Pull out the common factor 2x first
- 2x(x + 3)
Takeaway: Always pull out a common factor first.