Example 2
Completion requirements
| Example 2 |
Write the trinomial \(6m^2 -39m -21 \) as a product of its factors.
Step 1:
If possible, identify the GCF.
The GCF is \(3\).
\(6m^2 - 39m - 21 = 3(2m^2 - 13m - 7) \)
Step 2: Find a pair of numbers with a product of \(ac\) (\(2(–7) = –14\) in this case) and a sum of \(b\) (\(–13\) in this case).
Note the trinomial that must now be factored is \(2m^2 - 13m - 7 \). However, do not forget about the GCF of \(3\). It must be carried through as a factor to the very end.
Step 3: The numbers \(1\) and \(–14\) add to \(–13\) and multiply to \(–14\). Rewrite the middle term (\(b\)-term) as two terms with coefficients \(1\) and \(–14\).
\(2m^2 - 13m - 7 = 2m^2 + {\color{red}1m - 14m} - 7\)
Step 4: Group the first two terms and the last two terms.
\(2m^2 + 1m - 14m - 7 = (2m^2 + m) + (-14m - 7)\)
Step 5: Remove the greatest common factor from each of the two groups.
\((2m^2 + m) + (-14m - 7) = m(2m + 1) - 7(2m + 1)\)
Step 6: The result is a common factor, \(2m + 1\), in each group. Remove the common factor and simplify.
\(m({\color{red}2m + 1}) - 7({\color{red}2m + 1}) = ({\color{red}2m + 1})(m - 7)\)
You have now factored the trinomial into its three factors. Don't forget about the GCF of \(3\).
\(6m^2 -39m -21 = 3(2m + 1)(m - 7)\)
The GCF is \(3\).
\(6m^2 - 39m - 21 = 3(2m^2 - 13m - 7) \)
Step 2: Find a pair of numbers with a product of \(ac\) (\(2(–7) = –14\) in this case) and a sum of \(b\) (\(–13\) in this case).
Note the trinomial that must now be factored is \(2m^2 - 13m - 7 \). However, do not forget about the GCF of \(3\). It must be carried through as a factor to the very end.
| First number | Second number | Product | Sum | Works? |
| \(1\) | \(–14\) | \(–14\) | \(–13\) |
Yes!
|
| \(–1\) | \(14\) | \(–14\) | \(13\) | No |
| \(2\) | \(–7\) | \(–14\) | \(–5\) | No |
| \(–2\) | \(7\) | \(–14\) | \(5\) | No |
Step 3: The numbers \(1\) and \(–14\) add to \(–13\) and multiply to \(–14\). Rewrite the middle term (\(b\)-term) as two terms with coefficients \(1\) and \(–14\).
\(2m^2 - 13m - 7 = 2m^2 + {\color{red}1m - 14m} - 7\)
Step 4: Group the first two terms and the last two terms.
\(2m^2 + 1m - 14m - 7 = (2m^2 + m) + (-14m - 7)\)
Step 5: Remove the greatest common factor from each of the two groups.
\((2m^2 + m) + (-14m - 7) = m(2m + 1) - 7(2m + 1)\)
Step 6: The result is a common factor, \(2m + 1\), in each group. Remove the common factor and simplify.
\(m({\color{red}2m + 1}) - 7({\color{red}2m + 1}) = ({\color{red}2m + 1})(m - 7)\)
You have now factored the trinomial into its three factors. Don't forget about the GCF of \(3\).
\(6m^2 -39m -21 = 3(2m + 1)(m - 7)\)