/*! 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 4 Reduce each of the following fra... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 4

Reduce each of the following fractions to lowest terms. $$ \frac{-3 a^{4}+75 a^{2}}{2 a^{3}-16 a^{2}+30 a} $$

Short Answer

Expert verified
Question: Reduce the following fraction to its lowest terms: $$ \frac{-3 a^{4}+75 a^{2}}{2 a^{3}-16 a^{2}+30 a}. $$ Answer: The reduced fraction is: $$ \frac{3(a + 5)}{2(a - 3)}. $$

Step by step solution

01

Factor out common terms from numerator and denominator

In the given fraction, $$ \frac{-3 a^{4}+75 a^{2}}{2 a^{3}-16 a^{2}+30 a}, $$ we observe that the numerator has a common factor of \(3a^2\) and the denominator has a common factor of \(2a\). Let's factor these out: $$ \frac{3a^2(-a^{2}+25)}{2a(a^{2}-8a+15)}. $$
02

Factorize the numerator and the denominator further

We can factorize the numerator further by using the difference of squares formula: $$ (-a^2 + 25) = (a + 5)(-a + 5). $$ For the denominator, we can try to factorize the quadratic expression \(a^2-8a+15\): $$ (a^2 - 8a + 15) = (a - 3)(a - 5). $$ So now, our fraction looks like: $$ \frac{3a^2(a + 5)(-a + 5)}{2a(a-3)(a-5)}. $$
03

Cancel out any common factors

Let's cancel out the common factors from the numerator and the denominator of the fraction: $$ \require{cancel} \frac{3\cancel{a^2}(a + 5)(\cancel{-a + 5})}{2\cancel{a}(a-3)(\cancel{a-5})}. $$ After cancelling common factors, we are left with the reduced fraction: $$ \frac{3(a + 5)}{2(a - 3)}. $$

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!

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

For the following problems, divide the polynomials. $$ m^{4}+2 m^{3}-8 m^{2}-m+2 \text { by } m-2 $$

When the same number is subtracted from both the numerator and denominator of \(\frac{3}{10},\) the result is \(\frac{1}{8}\). What is the number that is subtracted?

For the following problems, perform the divisions. $$ \frac{8 z^{6}-4 z^{5}-8 z^{4}+8 z^{3}+3 z^{2}-14 z}{2 z-3} $$

For the following problems, perform the indicated operations. $$ \frac{x-3}{(x+2)(x+4)}+\frac{2 x-1}{x+4} $$

For the following problems, find the solution. Debbie can complete an algebra assignment in \(\frac{3}{4}\) of an hour. Sandi, who plays her radio while working, can complete the same assignment in \(1 \frac{1}{4}\) hours. If Debbie and Sandi work together, how long will it take them to complete the assignment?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

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.