Factorising Trinomials of the Form <em>ax</em><sup>2</sup> + <em>bx</em> + <em>c</em> With a GCF
Factorising Trinomials of the form ax2 + bx + c with a GCF
Now that we have organized what we’ve covered so far, we are ready to factor trinomials whose leading coefficient is not 1, trinomials of the form \(a{x}^{2}+bx+c\).
Remember to always check for a GCF first! Sometimes, after you factor the GCF, the leading coefficient of the trinomial becomes 1 and you can factor it by the methods in the last section. Let’s do a few examples to see how this works.
Watch out for the signs in the next two examples.
Example
Factor completely: \(2{n}^{2}-8n-42\).
Solution
Use the preliminary strategy.
\(\begin{array}{cccc}\text{Is there a greatest common factor?}\hfill & & & 2{n}^{2}-8n-42\hfill \\ \phantom{\rule{2.5em}{0ex}}\text{Yes, GCF = 2. Factor it out.}\hfill & & & 2\left({n}^{2}-4n-21\right)\hfill \\ \text{Inside the parentheses, is it a binomial, trinomial, or are there}\hfill & & & \\ \text{more than three terms?}\hfill & & & \\ \phantom{\rule{2.5em}{0ex}}\text{It is a trinomial whose coefficient is 1, so undo FOIL.}\hfill & & & 2\left(n\phantom{\rule{1em}{0ex}}\right)\left(n\phantom{\rule{1em}{0ex}}\right)\hfill \\ \phantom{\rule{2.5em}{0ex}}\text{Use 3 and}\phantom{\rule{0.2em}{0ex}}-7\phantom{\rule{0.2em}{0ex}}\text{as the last terms of the binomials.}\hfill & & & 2\left(n+3\right)\left(n-7\right)\hfill \end{array}\)
| Factors of \(-21\) | Sum of factors |
|---|---|
| \(1,-21\) | \(1+\left(-21\right)=-20\) |
| \(3,-7\) | \(3+\left(-7\right)=-4\text{*}\) |
Check.
\(\phantom{\rule{2.5em}{0ex}}2\left(n+3\right)\left(n-7\right)\)
\(\phantom{\rule{2.5em}{0ex}}2\left({n}^{2}-7n+3n-21\right)\)
\(\phantom{\rule{2.5em}{0ex}}2\left({n}^{2}-4n-21\right)\)
\(\phantom{\rule{2.5em}{0ex}}2{n}^{2}-8n-42\phantom{\rule{0.2em}{0ex}}✓\)
Example
Factor completely: \(4{y}^{2}-36y+56\).
Solution
Use the preliminary strategy.
\(\begin{array}{cccc}\text{Is there a greatest common factor?}\hfill & & & 4{y}^{2}-36y+56\hfill \\ \phantom{\rule{2.5em}{0ex}}\text{Yes, GCF = 4. Factor it.}\hfill & & & 4\left({y}^{2}-9y+14\right)\hfill \\ \text{Inside the parentheses, is it a binomial, trinomial, or are}\hfill & & & \\ \text{there more than three terms?}\hfill & & & \\ \phantom{\rule{2.5em}{0ex}}\text{It is a trinomial whose coefficient is 1. So undo FOIL.}\hfill & & & 4\left(y\phantom{\rule{1.5em}{0ex}}\right)\left(y\phantom{\rule{1.5em}{0ex}}\right)\hfill \\ \text{Use a table like the one below to find two numbers that multiply to}\hfill & & & \\ 14\phantom{\rule{0.2em}{0ex}}\text{and add to}\phantom{\rule{0.2em}{0ex}}-9.\hfill & & & \\ \text{Both factors of 14 must be negative.}\hfill & & & 4\left(y-2\right)\left(y-7\right)\hfill \end{array}\)
| Factors of \(14\) | Sum of factors |
|---|---|
| \(-1,-14\) | \(-1+\left(-14\right)=-15\) |
| \(-2,-7\) | \(-2+\left(-7\right)=-9\text{*}\) |
Check.
\(\phantom{\rule{2.5em}{0ex}}4\left(y-2\right)\left(y-7\right)\)
\(\phantom{\rule{2.5em}{0ex}}4\left({y}^{2}-7y-2y+14\right)\)
\(\phantom{\rule{2.5em}{0ex}}4\left({y}^{2}-9y+14\right)\)
\(\phantom{\rule{2.5em}{0ex}}4{y}^{2}-36y+42\phantom{\rule{0.2em}{0ex}}✓\)
In the next example the GCF will include a variable.
Example
Factor completely: \(4{u}^{3}+16{u}^{2}-20u\).
Solution
Use the preliminary strategy.
\(\begin{array}{cccc}\text{Is there a greatest common factor?}\hfill & & & 4{u}^{3}+16{u}^{2}-20u\hfill \\ \phantom{\rule{2.5em}{0ex}}\text{Yes, GCF = 4}u.\phantom{\rule{0.2em}{0ex}}\text{Factor it.}\hfill & & & 4u\left({u}^{2}+4u-5\right)\hfill \\ \text{Binomial, trinomial, or more than three terms?}\hfill & & & \\ \phantom{\rule{2.5em}{0ex}}\text{It is a trinomial. So “undo FOIL.”}\hfill & & & 4u\left(u\phantom{\rule{1em}{0ex}}\right)\left(u\phantom{\rule{1em}{0ex}}\right)\hfill \\ \text{Use a table like the table below to find two numbers that}\hfill & & & 4u\left(u-1\right)\left(u+5\right)\hfill \\ \text{multiply to}\phantom{\rule{0.2em}{0ex}}-5\phantom{\rule{0.2em}{0ex}}\text{and add to}\phantom{\rule{0.2em}{0ex}}4.\hfill & & & \end{array}\)
| Factors of \(-5\) | Sum of factors |
|---|---|
| \(-1,5\) | \(-1+5=4\text{*}\) |
| \(1,-5\) | \(1+\left(-5\right)=-4\) |
Check.
\(\phantom{\rule{2.5em}{0ex}}4u\left(u-1\right)\left(u+5\right)\)
\(\phantom{\rule{2.5em}{0ex}}4u\left({u}^{2}+5u-u-5\right)\)
\(\phantom{\rule{2.5em}{0ex}}4u\left({u}^{2}+4u-5\right)\)
\(\phantom{\rule{2.5em}{0ex}}4{u}^{3}+16{u}^{2}-20u\phantom{\rule{0.2em}{0ex}}✓\)
This lesson is part of:
Factoring and Factorisation I