/*! 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 3 Factor \(6 x^{3}-2 x^{2}\).... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 3

Factor \(6 x^{3}-2 x^{2}\).

Short Answer

Expert verified
The factored form is \(2x^2(3x - 1)\).

Step by step solution

01

Find the Greatest Common Factor (GCF)

Identify the greatest common factor of the coefficients and the variable terms. For the expression \(6x^3 - 2x^2\), the GCF of the coefficients 6 and 2 is 2. The common factor in the variable terms \(x^3\) and \(x^2\) is \(x^2\). Therefore, the GCF is \(2x^2\).
02

Factor out the GCF

Divide each term in the expression \(6x^3 - 2x^2\) by the GCF \(2x^2\). This gives: \[\frac{6x^3}{2x^2} = 3x\] and \[\frac{2x^2}{2x^2} = 1\]. Now, factor out \(2x^2\) from the original expression: \(6x^3 - 2x^2 = 2x^2(3x - 1)\).

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) is crucial when factoring polynomials.
The GCF is the largest number, including variables, that can divide each term without leaving a remainder.
For example, in the polynomial expression \(6x^3 - 2x^2\), it's essential to find the GCF of both the coefficients and the variable terms.
To find the GCF of the coefficients 6 and 2, you look at their greatest common divisor, which is 2.
Then, examine the variables \(x^3\) and \(x^2\). The common factor is \(x^2\). Therefore, the GCF is \(2x^2\).
Once you identify the GCF, you can factor it out to simplify the polynomial.
Polynomial Division
Polynomial division is the process of dividing each term in the polynomial by the GCF.
This step makes the polynomial easier to handle and understand.
Once we know the GCF \(2x^2\) for \(6x^3 - 2x^2\), we proceed by dividing each term:
\[ \frac{6x^3}{2x^2} = 3x \]
\[ \frac{2x^2}{2x^2} = 1 \]
By efficiently performing these divisions, we simplify the expression into its factored form.
This step helps break down the polynomial into smaller, more manageable terms.
Variable Terms
In polynomial expressions, variable terms denote the parts of the expression that contain variables like x, y, etc.
It鈥檚 vital to consider these variable terms when factoring polynomials.
Identifying the highest power of the variable common to each term is key in finding the GCF.
In the expression \(6x^3 - 2x^2\), both terms share the variable \(x\).
The lowest exponent of \(x\) common to both is \(x^2\).
This means \(x^2\) is part of the GCF, along with 2 (the GCD of the coefficients).
By factoring out \(2x^2\), we simplify the polynomial to:
\(6x^3 - 2x^2 = 2x^2(3x - 1)\).
Now, the polynomial is easier to work with.

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

The purity of gold is measured in karats, with pure gold being 24 karats. Other purities of gold are expressed as proportional parts of pure gold. Thus, 18-karat gold is \(\frac{18}{24},\) or \(75 \%\) pure gold; 12 -karat gold is \(\frac{12}{24},\) or \(50 \%\) pure gold; and so on. How much 12-karat gold should be mixed with pure gold to obtain 60 grams of 16-karat gold?

The period of a pendulum is the time it takes the pendulum to make one full swing back and forth. The period \(T,\) in seconds, is given by the formula \(T=2 \pi \sqrt{\frac{l}{32}}\) where \(l\) is the length, in feet, of the pendulum. In 1851 , Jean Bernard Leon Foucault win Paris. The period of Foucault's pendulum was approximately 16.5 seconds. What was its length?

Challenge Problem Harmonic Mean For \(0

In going from Chicago to Atlanta, a car averages 45 miles per hour, and in going from Atlanta to Miami, it averages 55 miles per hour. If Atlanta is halfway between Chicago and Miami, what is the average speed from Chicago to Miami? Discuss an intuitive solution. Write a paragraph defending your intuitive solution. Then solve the problem algebraically. Is your intuitive solution the same as the algebraic one? If not, find the flaw.

Arithmetic Mean If \(a
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.