Factoring Trinomials When the Leading Coefficient Is Not One Cheat Sheet

Key Facts

• A standard trinomial has the form $ax^2 + bx + c$, where $a \neq 0$.
• When $a \neq 1$, use the AC method by finding two numbers whose product is $a \cdot c$ and whose sum is $b$.
• After finding the two numbers, rewrite $bx$ as a sum such as $m x + n x$, where $m + n = b$ and $mn = a \cdot c$.
• Factor by grouping after splitting the middle term: group the first two terms and the last two terms.
• The factored form usually looks like $(px + q)(rx + s)$, where $pr = a$ and $qs = c$.
• If $c$ is positive, the two factor numbers have the same sign, and if $c$ is negative, they have opposite signs.
• If all terms share a greatest common factor, factor out the GCF before using the AC method.
• Check every factorization by multiplying the binomials with FOIL to confirm that the result is $ax^2 + bx + c$.

Vocabulary

Trinomial

A trinomial is a polynomial with three terms, such as $ax^2 + bx + c$.

Leading coefficient

The leading coefficient is the coefficient of the highest-degree term, such as $a$ in $ax^2 + bx + c$.

AC method

The AC method is a factoring strategy that uses the product $a \cdot c$ to split the middle term.

Greatest common factor

The greatest common factor is the largest factor shared by all terms in an expression.

Factoring by grouping

Factoring by grouping rewrites four terms as two groups that share a common binomial factor.

Binomial factor

A binomial factor is a two-term expression, such as $(2x + 3)$, that multiplies with another factor to make the original expression.