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Chapter 5: Problem 99

Factor by grouping. \(16 m^{3}-4 m^{2} p^{2}-4 m p+p^{3}\)

Short Answer

Expert verified
Factor: \( (4m - p^2)(4m^2 - p) \)

Step by step solution

01

Group Terms

Group the given polynomial into pairs: \( (16m^3 - 4m^2p^2) + (-4mp + p^3) \)
02

Factor Out the Greatest Common Factor (GCF) in Each Group

Factor out the GCF from each group: \( 4m^2(4m - p^2) - p(4m - p^2) \)
03

Identify and Factor Out the Common Binomial

Notice that \(4m - p^2\) is a common factor. Factor it out: \( (4m^2 - p)(4m - p^2) \)

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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 helps to simplify polynomials by breaking them into smaller, more manageable parts.
This technique is especially useful when you have four or more terms.
First, divide the polynomial into two pairs of terms.
For example, with the polynomial given in the exercise, you get:
\( (16m^3 - 4m^2p^2) + (-4mp + p^3) \)

Now, you have two groups: \( 16m^3 - 4m^2p^2 \) and \( -4mp + p^3 \).
This separation sets the stage for factoring out the Greatest Common Factor (GCF) in each group.
greatest common factor (GCF)
The Greatest Common Factor, or GCF, is the largest factor shared by all terms in a set.
Finding the GCF simplifies each group of terms.
In the first group \( 16m^3 - 4m^2p^2 \), you can factor out \( 4m^2 \), giving:
\( 4m^2(4m - p^2) \)
In the second group \( -4mp + p^3 \), you can factor out \( p \), resulting in:
\( -p(4m - p^2) \).

Now, the polynomial looks like this:
\( 4m^2(4m - p^2) - p(4m - p^2) \).
The GCF in each group makes the next step, identifying a common binomial factor, much easier.
binomial factorization
Binomial factorization involves finding a common binomial factor in different terms.
Once you identify the common binomial factor, you can factor it out.
For the polynomial, the common binomial factor in each group is \( 4m - p^2 \).
This allows you to rewrite the expression as:
\( (4m^2 - p)(4m - p^2) \).
Here, you see that the polynomial is now factored neatly into two binomials.

Understanding how to identify and factor out common binomial factors is crucial.
This technique simplifies the polynomial, making it easier to work with or solve.

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Most popular questions from this chapter

Solve each problem. Tram works due north of home. Her husband Alan works due east. They leave for work at the same time. By the time Tram is \(5 \mathrm{mi}\) from home, the distance between them is \(1 \mathrm{mi}\) more than Alan's distance from home. How far from home is Alan?

Factor each trinomial. \(-x^{2}+x+72\)

Which of the following are differences of cubes? A. \(9 x^{3}-125\) B. \(x^{3}-16\) C. \(x^{3}-1\) D. \(8 x^{3}-27 y^{3}\)

Factor each binomial completely. \(x^{3}-343\)

Solve each problem. Find three consecutive odd integers such that the sum of all three is 42 less than the product of the second and third integers.
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