Problem 81
Question
Factor by grouping. $$ 7 m^{3}-2 m^{2}+14 m-4 $$
Step-by-Step Solution
Verified Answer
\((m^2 + 2)(7m - 2)\)
1Step 1: Group Terms
First, we divide the polynomial into two groups to aim for a common factor within each group. Write the polynomial as:\[ (7m^3 - 2m^2) + (14m - 4) \]
2Step 2: Factor Out the Greatest Common Factor (GCF) from Each Group
Now, factor out the greatest common factor from each of these groups. For the first group \(7m^3 - 2m^2\), the GCF is \(m^2\), and for the second group \(14m - 4\), the GCF is \(2\). \[ m^2(7m - 2) + 2(7m - 2) \]
3Step 3: Identify and Factor Out the Common Binomial
Observe that both terms \(m^2(7m - 2)\) and \(2(7m - 2)\) include the common binomial \(7m - 2\). Factor \(7m - 2\) out of the expression:\[ (m^2 + 2)(7m - 2) \]
4Step 4: Write the Final Factored Expression
The entire polynomial has now been factored by grouping, resulting in:\[ (m^2 + 2)(7m - 2) \]
Key Concepts
Grouping MethodGreatest Common FactorBinomial Factorization
Grouping Method
The grouping method is a useful strategy for factoring polynomials, especially when a polynomial has four terms. The goal is to rearrange and group these terms in such a way that we can identify common factors easily. Let’s dive into how this method works step by step.
The first step is to divide the polynomial into two groups. With our example, we have the polynomial \(7m^3 - 2m^2 + 14m - 4\). We can group it into two parts: \((7m^3 - 2m^2)\) and \((14m - 4)\). Creating logical pairs or groups helps in simplifying the polynomial.
The key in grouping is to ensure that each group has a common factor that can be factored out in the subsequent steps. This step sets the foundation for easier factorization later on, as it simplifies the expression into two manageable parts that can be factored further.
The first step is to divide the polynomial into two groups. With our example, we have the polynomial \(7m^3 - 2m^2 + 14m - 4\). We can group it into two parts: \((7m^3 - 2m^2)\) and \((14m - 4)\). Creating logical pairs or groups helps in simplifying the polynomial.
The key in grouping is to ensure that each group has a common factor that can be factored out in the subsequent steps. This step sets the foundation for easier factorization later on, as it simplifies the expression into two manageable parts that can be factored further.
Greatest Common Factor
When factoring polynomials, finding the greatest common factor (GCF) is a crucial step. The GCF is the largest factor that divides each term in a group. For our exercise, once we have grouped the polynomial, we need to factor out the GCF from each group.
In the first group \(7m^3 - 2m^2\), the GCF is \(m^2\). This is because \(m^2\) is the largest common factor that exists in both terms \(7m^3\) and \(-2m^2\). Similarly, for the second group \(14m - 4\), the GCF is \(2\). Both of these numbers, \(14\) and \(-4\), can be divided by \(2\).
Factoring out the GCF results in simplified expressions \(m^2(7m - 2)\) and \(2(7m - 2)\). This process not only reduces complexity but also prepares the polynomial for the final step of binomial factorization.
In the first group \(7m^3 - 2m^2\), the GCF is \(m^2\). This is because \(m^2\) is the largest common factor that exists in both terms \(7m^3\) and \(-2m^2\). Similarly, for the second group \(14m - 4\), the GCF is \(2\). Both of these numbers, \(14\) and \(-4\), can be divided by \(2\).
Factoring out the GCF results in simplified expressions \(m^2(7m - 2)\) and \(2(7m - 2)\). This process not only reduces complexity but also prepares the polynomial for the final step of binomial factorization.
Binomial Factorization
After simplifying each group by factoring out the greatest common factor, the next task is to identify and extract common binomials, which leads to the final step of binomial factorization.
In our example, the expression \(m^2(7m - 2) + 2(7m - 2)\) is obtained from factoring out the GCF from each group. Notice that both parts contain the identical binomial \((7m - 2)\). This means we can factor \((7m - 2)\) out of the entire expression.
By recognizing this common binomial, we rewrite the polynomial as \((m^2 + 2)(7m - 2)\). This step shows the power of binomial factorization, where identifying identical binomials allows further simplification, efficiently breaking down the polynomial to its factored form. This technique helps in solving equations, finding roots, and simplifying expressions.
In our example, the expression \(m^2(7m - 2) + 2(7m - 2)\) is obtained from factoring out the GCF from each group. Notice that both parts contain the identical binomial \((7m - 2)\). This means we can factor \((7m - 2)\) out of the entire expression.
By recognizing this common binomial, we rewrite the polynomial as \((m^2 + 2)(7m - 2)\). This step shows the power of binomial factorization, where identifying identical binomials allows further simplification, efficiently breaking down the polynomial to its factored form. This technique helps in solving equations, finding roots, and simplifying expressions.
Other exercises in this chapter
Problem 81
Factor. $$ 64 x^{12}+y^{15} z^{18} $$
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Graph the interval \((-2,3]\) on a number line.
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Solve each equation. $$ 4 y^{2}-11 y-3=0 $$
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