Factoring Trinomials

To factor $ax^2+bx+c$, find binomials whose product recreates the leading term, middle term, and constant.

Start with the GCF

Before trinomial methods, factor any common factor. For

then factor inside to obtain $3(x+2)(x+3)$.

Monic trinomials

When $a=1$, seek numbers $p$ and $q$ such that

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

Sign reasoning

If $c>0$, the two numbers share a sign; use the sign of $b$. If $c<0$, signs differ; the sign of $b$ belongs to the number with larger magnitude.

This narrows the search before testing factor pairs.

Non-monic trinomials

For $ax^2+bx+c$ with $a\ne1$, find two numbers whose product is $ac$ and sum is $b$. Split the middle term and factor by grouping.

For $6x^2+11x+3$, $ac=18$ and $9+2=11$:

Systematic trial

You may also list factor pairs for $a$ and $c$, then check outer plus inner products. Keep the leading and constant products visible. Guessing without a system repeats work and hides sign logic.

Perfect-square trinomials

Patterns

and

occur when first and last terms are squares and the middle term is twice their product. For $x^2-10x+25$, factors are $(x-5)^2$.

Solving equations

Write zero on one side, factor, then apply zero-product property. From $2x^2-5x-3=0$,

so $x=-1/2$ or $x=3$.

When a trinomial is prime

Not every trinomial factors over integers. $x^2+x+1$ has no integer pair multiplying to $1$ and adding to $1$. Depending on the task, use the quadratic formula or leave it irreducible over the integers.

Verifying

Multiply factors using distribution. For $(3x+1)(2x+3)$, products $6x^2$, $9x$, $2x$, and $3$ combine to $6x^2+11x+3$.

Common mistakes

Ignoring a GCF. Factor it first.

Using numbers that multiply correctly but add incorrectly. Both conditions matter.

Forgetting the leading coefficient in $ac$. Non-monic trinomials need extra structure.

Assuming every trinomial factors over integers. Some require other methods.

Quick self-check

• Is the GCF removed?
• What are $a$, $b$, $c$, and $ac$?
• Do the chosen numbers have correct product and sum?
• Does expansion reproduce the original trinomial?

Related topics

• Factoring
• Greatest Common Factor Factoring
• Perfect Square Trinomials
• Quadratic Equation