Key Concepts

Factoring Trinomials in the form [latex]ax^{2}+bx+c[/latex]

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

[latex]\begin{array}{l}r\cdot{s}=a\cdot{c}\\r+s=b\end{array}[/latex]

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

How to factor a trinomial in the form [latex]a{x}^{2}+bx+c[/latex] by grouping

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

Factoring Trinomials in the form [latex]x^{2}+bx+c[/latex]

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

[latex]\begin{array}{l}r\cdot{s}=c\\\text{ and }\\r+s=b\end{array}[/latex]

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

How to factor a trinomial in the form [latex]{x}^{2}+bx+c[/latex]

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

Glossary

Prime trinomial – A trinomial that cannot be factored using integers