/*! 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 69 Factor. $$ 18 x y^{3}+3 x y^... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 69

Factor. $$ 18 x y^{3}+3 x y^{2}-10 x y $$

Short Answer

Expert verified
The factored form is \(3xy(6y^2 + y - \frac{10}{3})\).

Step by step solution

01

- Factor out the Greatest Common Factor (GCF)

Identify and factor out the greatest common factor from all terms. In this expression, the GCF for all terms is \(3xy\). Divide each term by \(3xy\) and rewrite the expression.o \(\frac{18xy^{3}}{3xy} = 6y^2$$\frac{3xy^2}{3xy} = y$$\frac{ -10xy}{3xy} = -\frac{10}{3}\)So the factored expression becomes: \(3xy(6y^2 + y - \frac{10}{3})\).
02

- Simplify the Remaining Expression

Combine like terms and simplify the factored form. In our case, the expression within the parentheses doesn鈥檛 appear to factor further over the integers, so our final expression is left as:\(3xy(6y^2 + y - \frac{10}{3})\).

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.

Finding the Greatest Common Factor (GCF)
The first step in factoring any algebraic expression is to determine the greatest common factor (GCF). This is the largest factor that divides all terms in the expression.
Identifying the GCF involves looking at both the coefficients and the variables in each term.
For example, in the expression \[18xy^3 + 3xy^2 - 10xy\], all terms share the common factor of \(3xy\). By factoring out the \(3xy\), we can simplify the expression easily.

Here鈥檚 how it鈥檚 done:
  • Divide each term by \(3xy\).
  • This gives us:
    • \[ \frac{18xy^3}{3xy} = 6y^2 \ \frac{3xy^2}{3xy} = y \ \frac{-10xy}{3xy} = -\frac{10}{3} \]
  • So, the factored form is \(3xy(6y^2 + y - \frac{10}{3})\).
By factoring out the GCF, we make the expression simpler and easier to work with, laying the groundwork for further simplification.
Factoring Polynomials
Once the GCF is factored out, the next step is to try and factor the remaining polynomial.
Factoring polynomials involves breaking down the polynomial into a product of simpler polynomials. However, not all polynomials can be factored further using integers.

In the example expression \(3xy(6y^2 + y - \frac{10}{3})\):
  • The polynomial inside the parentheses, \(6y^2 + y - \frac{10}{3}\), needs to be checked to see if it can be factored further.
  • Often, we look for factors that multiply to give the constant term (in this case, \(-\frac{10}{3}\)) and add up to the coefficient of the middle term \(1\).
  • In this specific instance, the polynomial \(6y^2 + y - \frac{10}{3}\) does not factor neatly over the integers.
Therefore, we consider the factorization process complete at this stage. Learning to recognize when a polynomial can or cannot be factored further is an important skill in algebra.
Simplifying Expressions
Simplifying refers to reducing an expression to its simplest form, which often involves combining like terms or reducing fractions.

For the expression \(3xy(6y^2 + y - \frac{10}{3})\), we check for any additional simplifications that can be made:
  • Look for like terms that can be combined.
  • In our case, \(6y^2\), \(y\), and \(-\frac{10}{3}\) are already in their simplest polynomial form.
  • No further combining or simplification within the parentheses can be done over the integers.
Therefore, \(3xy(6y^2 + y - \frac{10}{3})\) remains as the final simplified form.

Understanding simplifying expressions is essential for working with more complex algebraic problems. By breaking down each step, you ensure accuracy and a more manageable expression.

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

Factor completely. $$ (y-1)^{4}-(y-1)^{2} $$

Find the slope and the \(y\) -intercept of the graph of \(2 x-3 y=6\)

Factor completely. $$ -16+17\left(5-y^{2}\right)-\left(5-y^{2}\right)^{2} $$

Factor completely. $$ 24 t^{2 a}-6 $$

Use a graphing calculator to find any real-number solutions that exist accurate to two decimal places. $$x^{3}-3.48 x^{2}+x=3.48$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

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.