Problem 76 Factor each four-term polynomial... [FREE SOLUTION]

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Chapter 6: Problem 76

Factor each four-term polynomial by grouping. If this is not possible, write "not factorable by grouping." $$ 6 m^{2}-5 m n-6 m+5 n $$

Short Answer

Expert verified

Factored as $(6m - 5n)(m - 1)$.

Step by step solution

Group the First Two Terms

Take the polynomial $6m^2 - 5mn - 6m + 5n$ and group the first two terms. This gives us $(6m^2 - 5mn)$.

Group the Last Two Terms

Now, group the last two terms of the polynomial, resulting in $(-6m + 5n)$.

Factor Common Factors from Each Group

For the first group $(6m^2 - 5mn)$, factor out the common factor, which is $m$: $m(6m - 5n)$. For the second group $(-6m + 5n)$, factor out $-1$ to get $-1(6m - 5n)$.

Identify Common Binomial Factor

Notice now that each group has $(6m - 5n)$ as a common factor: $m(6m - 5n) - 1(6m - 5n)$.

Factor Out the Common Binomial

The expression can now be written as $(6m - 5n)(m - 1)$ by factoring out the common binomial $(6m - 5n)$. This is the completely factored form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring by Grouping

Factoring by grouping is a useful method to factor polynomials, especially those with four terms. The objective is to rearrange and group the terms in such a way that each group has a common factor, which can then be factored out. To apply this technique:

Step 1: Identify and group terms. For a four-term polynomial, divide it into two pairs or groups of terms.
Step 2: Factor each group separately by looking for common factors.
Step 3: If both groups have a common binomial factor, factor this binomial out to simplify the expression.

Typically, if grouping doesn't reveal a common factor, then the polynomial might not be factorable by this method. This approach excels when a polynomial can be broken down into recognizable patterns or simpler products.

Binomial Factor

When factoring polynomials by grouping, the goal often is to identify a common binomial factor. A binomial is an algebraic expression with two terms, such as (6m - 5n) in our example. Here鈥檚 why identifying a binomial factor is important:

It simplifies polynomial expressions to smaller components.
All terms in the polynomial can sometimes be expressed in terms of this binomial factor.
Rightly identifying a common binomial can easily resolve the polynomial into simpler factors.

Once you have a common factor like (6m - 5n), the next step is essential: factor it out. The remaining expression, (m - 1) for instance, reveals the entire structure of the polynomial. This process is key to simplification and further algebraic manipulation.

Common Factors in Polynomials

Common factors play a vital role in polynomial factoring. They are the numbers or terms that are common across multiple groups within the polynomial. Here's how to find and use them effectively:

Identify: Look for terms that repeat across different parts of the polynomial expression.
Factor Out: Pull out these common factors from individual groups to simplify the polynomial.
Simplify: Once common factors are extracted, the polynomial breaks down into easier components.

Finding common factors requires a careful examination of the terms and often begins with inspecting coefficients and variables. In our example, noticing that m and -1 both link up with the binomial (6m - 5n) is crucial. Factoring them out results in a neater and more manageable expression.

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91影视

Factor each four-term polynomial by grouping. If this is not possible, write "not factorable by grouping." $$ 6 m^{2}-5 m n-6 m+5 n $$

Short Answer

Step by step solution

Group the First Two Terms

Group the Last Two Terms

Factor Common Factors from Each Group

Identify Common Binomial Factor

Factor Out the Common Binomial

Key Concepts

Factoring by Grouping

Binomial Factor

Common Factors in Polynomials

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