Factor Trinomials With Leading Coefficient One

Factoring a trinomial with leading coefficient 1

A trinomial with leading coefficient 1 has the form [ x^2+bx+c. ] The goal is to rewrite it as a product of two binomials: [ (x+m)(x+n). ]

1) Find the two numbers

Look for two numbers that:

• multiply to c
• add to b

These numbers will become m and n.

2) Write the factorization

Once you find the pair, write [ x^2+bx+c=(x+m)(x+n). ] Be careful with signs: a positive product with a negative sum means both numbers are negative; a negative product means one number is positive and the other is negative.

3) Check your result

Multiply the binomials to make sure you get back the original trinomial. Expanding should give the middle term and constant term exactly.

Quick example pattern

If you have (x^2+7x+12), look for numbers that multiply to 12 and add to 7: 3 and 4. So the factorization is [ (x+3)(x+4). ]

Common check

If no pair seems to work, recheck the signs and all factor pairs of the constant term. The correct factorization should expand to the original trinomial without changing any coefficient.