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Chapter 6: Problem 54

Factor completely. Identify any prime polynomials. $$ 4 p^{2} z-20 p^{2}+p w z-5 p w $$

Short Answer

Expert verified
The factored form is \( (z - 5)(4p^2 + pw) \). Both factors are prime.

Step by step solution

01

- Group terms

Group the terms in pairs to make it easier to factor by grouping: \( (4p^2z - 20p^2) + (pwz - 5pw) \)
02

- Factor out the greatest common factor from each group

From the first group, factor out \(4p^2\). From the second group, factor out \(pw\): \( 4p^2(z - 5) + pw(z - 5) \)
03

- Factor out the common binomial factor

Both groups have a common binomial factor \( (z - 5) \). Factor this out: \( (z - 5)(4p^2 + pw) \)
04

- Identify prime polynomials

Check if each factor is already in its simplest form. The binomial \( (z - 5) \) and the polynomial \( (4p^2 + pw) \) are both prime.

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Key Concepts

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

Grouping Terms
Grouping terms is an essential initial step in factoring complex polynomials. Imagine you have a polynomial that looks a bit overwhelming. The trick here is to simplify it by grouping terms that share common factors. For example, in the polynomial given in the exercise:
(4p^2z - 20p^2) + (pwz - 5pw).
By grouping these terms into pairs, we create smaller, more manageable expressions. We can then focus on factoring each pair separately, making the entire process more straightforward.
Greatest Common Factor
Once you have grouped the terms, the next step is to factor out the Greatest Common Factor (GCF) from each group. The GCF is the largest factor that divides all terms in the group. It simplifies the polynomial and makes the next steps easier.
Let's see this in action: From the first group, (4p^2z - 20p^2), the GCF is 4p^2. By factoring it out, we get: 4p^2(z - 5). From the second group, (pwz - 5pw), the GCF is pw. By factoring it out, we get: pw(z - 5). Once the GCF is factored out, we can see common binomials more easily, which is crucial for the next step.
Prime Polynomials
After factoring out the GCF, we need to check if each polynomial factor is prime. A prime polynomial is one that cannot be factored further over the set of integers. In our exercise, we end up with: (z - 5)(4p^2 + pw).
To determine if these factors are prime, we check if they can be broken down more.
In this case, both (z - 5) and (4p^2 + pw) are already in their simplest forms and cannot be factored further, making them prime polynomials. Recognizing prime polynomials ensures we have fully simplified our original expression.

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For exercises 97-100, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Solve: \(x^{2}+12 x=0\) Incorrect Answer: \(x^{2}+12 x=0\) $$ \begin{aligned} \frac{x^{2}}{x}+\frac{12 x}{x} &=\frac{0}{x} \\ x+12 &=0 \\ \frac{-12}{x} &-12 \\ \hline x &=-12 \end{aligned} $$ Solution: \(x=-12\)
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