A trinomial is an expression of the form: \(ax^2+bx+c\).
Factoring trinomials finds the two binomial factors needed to write the expression in factored form: \(ax^2+bx+c=\left(r_1x+s_1\right)\left(r_2x+s_2\right)\).
Here is a simple example: \(x^2+7x+10=\left(x+2\right)\left(x+5\right)\). The question is how did we find the binomial factors?
The table below outlines the general procedure shown with the trinomial: \(x^2+7x+10\)
| Step 1: Multiply \(a\) and \(c\): | Here \(a=1\) and \(c=10\implies a\cdot c=10\) |
| Step 2: Find factors of \(a\cdot c\) that add up to \(b\): | \(b=7\) and \(10=2\cdot 5\implies 2+5=7\), so factors are \(2\) and \(5\) |
| Step 3: Rewrite trinomial into groups using factors: | \(\begin{array}{c} ax^2+rx \\ sx+c \end{array} \implies \begin{array}{c} 1x^2+2x \\ 5x+10 \end{array}\) |
| Step 4: Find the GCF of each group: | \(\begin{array}{ccc} 1x^2+2x & \implies & x\left(x+2\right) \\ 5x+10 & \implies & 5\left(x+2\right) \end{array}\) |
| Step 5: Factor out GCF: | GCF \(=\left(x+2\right)\implies \left(x+2\right)\left(x+5\right)\) |
Factor \(6x^2-7x-20\).
| Multiply \(a\cdot c\): | \(6\cdot -20=-120\) |
| Factors that add to \(b\): | \( -15\cdot 8=-120\) and \(-15x+8x=-7x\) |
| Rewrite in groups and GCFs: | \( \begin{array}{ccc} 6x^2-15x & \implies & 3x\left(2x-5\right) \\ 8x-20 & \implies & 4\left(2x-5\right) \end{array}\) |
| Write in factored form: | \(\left(2x-5\right)\left(3x+4\right)\;\color{green}{\checkmark}\) |
Factor \(18w^2-57w+35\).
| Multiply \(a\cdot c\): | \(18\cdot 35 = 630\) |
| Factors that add to \(b\): | \( -15\cdot -42 = 630\) and \(-15w-42wx=-57w\) |
| Rewrite in groups and GCFs: | \( \begin{array}{ccc} 18w^2-15w & \implies & 3w\left(6w-5\right) \\ -42w+35 & \implies & -7\left(6w-5\right) \end{array}\) |
| Write in factored form: | \(\left(6w-5\right)\left(3w-7\right)\;\color{green}{\checkmark}\) |
Watch the video I created below for an example on a different take on factoring than the one presented above. It might be more helpful for your understanding.
Watch for GCFs. Factor the trinomial: \(3v^2-24v+36\).
| GCF \(=3\), factor out of all terms: | \(\implies\) | \(3(v^2-8v+12)\) |
| Factor remaining trinomial: | \(\implies\) | Factors of \(12\) that add up to \(-8\) are \(-6\) and \(-2\) |
| \(\implies\) | \(3\left(v-2\right)\left(v-6\right)\;\color{green}{\checkmark}\) |