/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 Write in factored form by factor... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 39

Write in factored form by factoring out the greatest common factor. \(16 z^{4}+24 z^{2}\)

Short Answer

Expert verified
8z^2(2z^2 + 3)

Step by step solution

01

- Identify the Greatest Common Factor (GCF)

First, identify the greatest common factor of the given terms. The terms are 16z^4 and 24z^2. The coefficients are 16 and 24, and the variables are z^4 and z^2. The GCF of 16 and 24 is 8. The GCF of z^4 and z^2 is z^2. Therefore, the GCF of 16z^4 and 24z^2 is 8z^2.
02

- Factor out the GCF

Next, factor out the GCF from each term. Divide each term by the GCF (8z^2) and write the expression in factored form: \[ \frac{16z^4}{8z^2} + \frac{24z^2}{8z^2} = 2z^2 + 3 \].
03

- Write the Factored Form

Finally, write the original expression as the product of the GCF and the simplified expression from step 2: \[ 8z^2(2z^2 + 3) \].

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Greatest Common Factor
The greatest common factor (GCF) is crucial in simplifying polynomials. When you look at a polynomial like \(16z^4 + 24z^2\), you need to find the GCF of both the coefficients and the variable parts. Here, the coefficients are 16 and 24. The GCF of 16 and 24 is 8. For the variables, you have \(z^4\) and \(z^2\). The largest power of \(z\) that is common in both terms is \(z^2\). So, the GCF of the polynomial \(16z^4 + 24z^2\) is \(8z^2\). This simplifies the polynomial and makes it easier to factor out and solve problems.
Factored Form
Writing an expression in its factored form involves using the GCF you found. For our exercise, we found the GCF to be \(8z^2\). To write \(16z^4 + 24z^2\) in factored form, you need to factor \(8z^2\) out of each term.
Here's how it looks step by step:
  • Divide each term by the GCF \(8z^2\):
    \[ \frac{16z^4}{8z^2} + \frac{24z^2}{8z^2} \]
  • Simplify inside the parentheses: \[2z^2 + 3\]
  • Write the original polynomial as the product: \[8z^2(2z^2 + 3)\]
This shows each term divided by the GCF, resulting in a much simpler polynomial.
Polynomial Division
Polynomial division is often necessary when factoring. It involves dividing each term in the polynomial by the GCF. For our example, we had: \[16z^4 + 24z^2\]When we divide by \(8z^2\):
  • \[ \frac{16z^4}{8z^2} = 2z^2\]
  • \[ \frac{24z^2}{8z^2} = 3 \]
This division simplifies the polynomial.

Understanding polynomial division helps recognize patterns and solve higher-level problems. Practice this skill to become proficient in simplifying and factoring more complex polynomials.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

list all pairs of integers with the given product. Then find the pair whose sum is given. Product: \(-24 ; \quad\) Sum: \(-5\)

Factor by grouping. \(6-3 x-2 y+x y\)

If a trinomial has a negative coefficient for the squared term, as in \(-2 x^{2}+11 x-12,\) it is usually easier to factor by first factoring out the common factor \(-1 .\) $$ \begin{aligned} -2 x^{2}+11 x-12 \\ =&-1\left(2 x^{2}-11 x+12\right) \\ =&-1(2 x-3)(x-4) \end{aligned} $$ Use this method to factor each trinomial. See Example 7(b). $$ $$ -2 a^{2}-5 a b-2 b^{2} $$

Find the greatest common factor for each list of terms. \(16 y, 24\)

Factor completely. $$ t^{2}-t z-6 z^{2} $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.