/*! 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 52 In the following exercises, fact... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 52

In the following exercises, factor. $$ -4 x^{3} y^{5}-x^{2} y^{3}+12 x y^{4} $$

Short Answer

Expert verified
xy^3(-4x^2y^2 - x + 12y)

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Examine each term in the expression ewline -4x^3y^5, -x^2y^3, and 12xy^4 to find the greatest common factor (GCF). Each term has at least one x and at least three y's. The GCF is xy^3.
02

Factor out the GCF

Divide each term by xy^3 and factor it out. ewline -4x^3y^5 / xy^3 = -4x^2y^2, -x^2y^3 / xy^3 = -x, 12xy^4 / xy^3 = 12y. So, xy^3(-4x^2y^2 - x + 12y).
03

Rewrite the expression

Combine the results from step 2 to rewrite the original expression with the factored GCF: ewline xy^3(-4x^2y^2 - x + 12y).

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 (GCF)
The Greatest Common Factor (GCF) of a polynomial is the highest degree term that can be evenly divided into each term of the polynomial. In simple words, it鈥檚 the largest term that all parts of the equation share.
To find the GCF, you should:
  • Look at the coefficients and identify the largest number that all coefficients are divisible by.
  • Examine the variables and select the smallest powers that are common to all terms.

For example, in the polynomial \(-4x^3y^5 - x^2y^3 + 12xy^4\), each term shares at least one \(x\) and 3 \(y\)'s. Therefore, the GCF is \(xy^3\).
Polynomial Factoring
Polynomial factoring is the process of breaking down a polynomial into simpler terms (or factors) that can be multiplied together to get the original polynomial. Factoring polynomials is essential because it simplifies expressions, solving polynomial equations easier, and helps understand the properties of the polynomial.
To factor a given polynomial:
  • Identify the GCF.
  • Factor out the GCF from each term in the polynomial.
  • Rewrite the polynomial as a product of the GCF and the resulting simplified polynomial.

Using the provided example, for the polynomial \(-4x^3y^5 - x^2y^3 + 12xy^4\), after finding the GCF \((xy^3)\), you factor it out to get \(-4x^2y^2 - x + 12y\). Hence, the factored form is \(xy^3(-4x^2y^2 - x + 12y)\).
Algebraic Expression
An algebraic expression is a mathematical statement that involves variables, coefficients, and operations (such as addition, subtraction, multiplication, and division). Understanding algebraic expressions is crucial for algebra, as they form the basis for many algebraic procedures, including factoring.

Key points for working with algebraic expressions:
  • Recognize and combine like terms.
  • Apply the distributive property as needed.
  • Use factoring techniques to simplify expressions.

For instance, in the given expression \(-4x^3y^5 - x^2y^3 + 12xy^4\), the process of identifying the GCF and factoring it out is an application of understanding algebraic expressions. It allows us to simplify and rewrite the expression in a more manageable form, \(xy^3(-4x^2y^2 - x + 12y)\).

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

In the following exercises, factor each expression using any method. $$ 12 y^{2}-29 y+14 $$

In the following exercises, factor each expression using any method. $$ 5 r^{2}+25 r+30 $$

In the following exercises, factor using the 'ac' method. $$ 6 p^{2}+p-22 $$

In the following exercises, factor each trinomial of the form \(x^{2}+b x y+c y^{2} .\) $$ x^{2}-2 x y-80 y^{2} $$

In the following exercises, factor each trinomial of the form \(x^{2}+b x+c .\) $$ a^{2}+25 a+100 $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

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.