Problem 50
Question
Factor by grouping. \(m^{2}+6 m-12 m-72\)
Step-by-Step Solution
Verified Answer
(m + 6)(m - 12)
1Step 1 - Group the Terms
Group the terms in pairs in a way that makes it easier to factor. For the expression, group it as (m^{2} + 6m) - (12m + 72).
2Step 2 - Factor Out the Common Factor in Each Group
Factor out the common factor from each group separately. For the first group, the common factor is m. For the second group, the common factor is 12. This gives: m(m + 6) - 12(m + 6).
3Step 3 - Factor Out the Common Binomial
Now, both groups contain the common binomial (m + 6). Factor (m + 6) out of the expression: (m + 6)(m - 12).
4Step 4 - Verify the Factoring
Distribute (m + 6) to (m - 12) and check if the original expression is obtained: (m + 6)(m - 12) = m^{2} + 6m - 12m - 72. The original expression is correct, hence the factoring is verified.
Key Concepts
Grouping MethodCommon FactorBinomials
Grouping Method
The grouping method is a powerful technique for factoring algebraic expressions, especially useful when dealing with polynomials that have four or more terms. This method involves the following steps:
- First, group the terms in pairs in such a way that common factors can be easily found. For example, with the expression \(m^{2} + 6m - 12m - 72\), you can group it as \((m^{2} + 6m) - (12m + 72)\).
- Second, factor out the common factor from each of these groups. In the first group, \(m\) is a common factor, so we get \(m(m + 6)\). In the second group, \(12\) is the common factor, giving us \(-12(m + 6)\).
- Finally, you will notice that both groups now contain a common binomial factor, in this case, \((m + 6)\). You can then factor this binomial out, ending with \((m + 6)(m - 12)\).
Common Factor
Identifying a common factor is crucial when factoring algebraic expressions. A common factor is a term (which could be a number, variable, or a combination) that divides each term in a group evenly.
In the expression \(m^{2} + 6m - 12m - 72\), we look at the grouped terms: \((m^{2} + 6m)\) and \( - (12m + 72)\).
For \(m^{2} + 6m\), the common factor is \(m\) because \(m\) divides both \(m^{2}\) and \(6m\) evenly. For \( - 12m + 72\), the common factor is \(-12\) because \(-12\) divides both \(-12m\) and \(-72\) evenly.
Factoring out these common factors, we get \(m(m + 6) - 12(m + 6)\). Notice both groups now include the common binomial \( (m + 6) \). This step sets us up to then factor out the common binomial.
In the expression \(m^{2} + 6m - 12m - 72\), we look at the grouped terms: \((m^{2} + 6m)\) and \( - (12m + 72)\).
For \(m^{2} + 6m\), the common factor is \(m\) because \(m\) divides both \(m^{2}\) and \(6m\) evenly. For \( - 12m + 72\), the common factor is \(-12\) because \(-12\) divides both \(-12m\) and \(-72\) evenly.
Factoring out these common factors, we get \(m(m + 6) - 12(m + 6)\). Notice both groups now include the common binomial \( (m + 6) \). This step sets us up to then factor out the common binomial.
Binomials
Binomials are algebraic expressions containing exactly two terms. Recognizing binomials is important, as many factoring techniques, especially the grouping method, rely on identifying and factoring binomials.
In our example, after factoring out the common factors from grouped terms, we have a situation like this: \(m(m + 6) - 12(m + 6)\).
Here, \(m + 6\) is a binomial common to both terms. We can further factor the expression by treating \(m + 6\) as a single entity.
Factoring out the binomial, we get \((m + 6)(m - 12)\). Each term in the original expression contributes to forming this product of binomials. Verifying our result, distributed back, yields the original polynomial: \((m + 6)(m - 12) = m^{2} + 6m - 12m - 72\). Mastery of binomials helps tackle similar problems with confidence.
In our example, after factoring out the common factors from grouped terms, we have a situation like this: \(m(m + 6) - 12(m + 6)\).
Here, \(m + 6\) is a binomial common to both terms. We can further factor the expression by treating \(m + 6\) as a single entity.
Factoring out the binomial, we get \((m + 6)(m - 12)\). Each term in the original expression contributes to forming this product of binomials. Verifying our result, distributed back, yields the original polynomial: \((m + 6)(m - 12) = m^{2} + 6m - 12m - 72\). Mastery of binomials helps tackle similar problems with confidence.