Key Concepts

Factoring Trinomials in the form $ax^{2}+bx+c$

To factor a trinomial in the form $ax^{2}+bx+c$, find two integers, r and s, whose sum is b and whose product is ac.

$\begin{array}{l}r\cdot{s}=a\cdot{c}\\r+s=b\end{array}$

Rewrite the trinomial as $ax^{2}+rx+sx+c$ and then use grouping and the distributive property to factor the polynomial.

How to factor a trinomial in the form $a{x}^{2}+bx+c$ by grouping

1. List factors of $ac$.
2. Find $p$ and $q$, a pair of factors of $ac$ with a sum of $b$.
3. Rewrite the original expression as $a{x}^{2}+px+qx+c$.
4. Pull out the GCF of $a{x}^{2}+px$.
5. Pull out the GCF of $qx+c$.
6. Factor out the GCF of the expression.

Factoring Trinomials in the form $x^{2}+bx+c$

To factor a trinomial in the form $x^{2}+bx+c$, find two integers, r and s, whose product is c and whose sum is b.

$\begin{array}{l}r\cdot{s}=c\\\text{ and }\\r+s=b\end{array}$

Rewrite the trinomial as $x^{2}+rx+sx+c$ and then use grouping and the distributive property to factor the polynomial. The resulting factors will be $\left(x+r\right)$ and $\left(x+s\right)$.

How to factor a trinomial in the form ${x}^{2}+bx+c$

1. List factors of $c$.
2. Find $p$ and $q$, a pair of factors of $c$ with a sum of $b$.
3. Write the factored expression $\left(x+p\right)\left(x+q\right)$.

Glossary

Prime trinomial – A trinomial that cannot be factored using integers