More on Factoring General Polynomials
Factor 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 \(a x^{2}+b x+c\).
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 7.30
Factor completely: \(2 n^{2}-8 n-42\).
Solution
Use the preliminary strategy.
|
Is there a greatest common factor? Yes, GCF = 2. Factor it out. |
\(2 n^{2}-8 n-42\) \(2\left(n^{2}-4 n-21\right)\) |
|
Inside the parentheses, is it a binomial, trinomial, or are there more than three terms? It is a trinomial whose coefficient is 1, so undo FOIL. Use 3 and −7 as the last terms of the binomials. |
\(2(n)(n)\) \(2(n+3)(n-7)\) |
| Factors of −21 | Sum of factors |
|---|---|
| \(1, −21\) | \(1+(−21)=−20\) |
| \(3,−7\) | \(3+(−7)=−4\ *\) |
Check.
\(2(n+3)(n-7)\)
\(2\left(n^{2}-7 n+3 n-21\right)\)
\(2\left(n^{2}-4 n-21\right)\)
\(2 n^{2}-8 n-42 \text{✓}\)
Try It 7.59
Factor completely: \(4 m^{2}-4 m-8\).
Try It 7.60
Factor completely: \(5 k^{2}-15 k-50\).
Example 7.31
Factor completely: \(4 y^{2}-36 y+56\).
Solution
Use the preliminary strategy.
|
Is there a greatest common factor? Yes, GCF = 4. Factor it. |
\(4 y^{2}-36 y+56\) \(4\left(y^{2}-9 y+14\right)\) |
|
Inside the parentheses, is it a binomial, trinomial, or are there more than three terms? It is a trinomial whose coefficient is 1. So undo FOIL. |
\(4(y \qquad)(y \qquad)\) |
| Use a table like the one below to find two numbers that multiply to 14 and add to −9. | |
| Both factors of 14 must be negative. | \(4(y-2)(y-7)\) |
| Factors of 14 | Sum of factors |
|---|---|
| \(−1,−14\) | \(−1+(−14)=−15\) |
| \(−2,−7\) | \(−2+(−7)=−9*\) |
Check.
\(4(y-2)(y-7)\)
\(4\left(y^{2}-7 y-2 y+14\right)\)
\(4\left(y^{2}-9 y+14 \text{✓}\right)\)
Try It 7.61
Factor completely: \(3 r^{2}-9 r+6\).
Try It 7.62
Factor completely: \(2 t^{2}-10 t+12\).
In the next Example the GCF will include a variable.
Example 7.32
Factor completely: \(4 u^{3}+16 u^{2}-20 u\).
Solution
Use the preliminary strategy.
|
Is there a greatest common factor? Yes, GCF = 4u. Factor it. |
\(4 u^{3}+16 u^{2}-20 u\) \(4 u\left(u^{2}+4 u-5\right)\) |
|
Binomial, trinomial, or more than three terms? It is a trinomial. So "undo FOIL". |
\(4 u(u)(u)\) |
| Use a table like the table below to find two numbers that multiply to -5 and add to 4. | \(4 u(u-1)(u+5)\) |
| Factors of −5 | Sum of factors |
|---|---|
| \(−1,5\) | \(−1+5=4*\) |
| \(1,−5\) | \(1+(−5)=−4\) |
Check.
\(4 u(u-1)(u+5)\)
\(4 u\left(u^{2}+5 u-u-5\right)\)
\(4 u\left(u^{2}+4 u-5\right)\)
\(4 u^{3}+16 u^{2}-20 u\text{✓}\)
Try It 7.63
Factor completely: \(5 x^{3}+15 x^{2}-20 x\).
Try It 7.64
Factor completely: \(6 y^{3}+18 y^{2}-60 y\).