/*! 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 80 Factor completely, or state that... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 80

Factor completely, or state that the polynomial is prime. $$x^{3}+2 x^{2}-x-2$$

Short Answer

Expert verified
The factored form of the given polynomial is \((x+2)(x - 1)(x + 1)\)

Step by step solution

01

Identify and Apply Factoring Pattern

The given polynomial is \(x^{3} + 2x^{2} - x - 2\). Group the terms to make it easier to figure out the factors: \((x^{3} + 2x^{2}) - (x + 2)\). This allows us to factor out common terms.
02

Factor by Grouping

Factor out the common terms from each group. From the first part, we can factor out \(x^{2}\) to get: \(x^{2}(x + 2)\). From the second group, 1 can be factored out, although it doesn't change the terms, so it remains as \(- (x + 2)\). This results in the factorized expression: \(x^{2}(x + 2) - 1 * (x + 2)\). Notice that there's a common binomial factor, which indicates that we can further factor the polynomial.
03

Further Factorization

With the common binomial factor identified, it can be factored out: \((x+2)(x^{2}-1)\). Now, the quadratic binomial can be further factored as difference of squares: \((x^{2} - 1) = (x - 1)(x + 1)\)
04

Final Result

Substituting the factored binomial back to our equation gives us the completely factorized form of the initial cubic function: \((x+2)(x - 1)(x + 1)\). Thus, the polynomial has been completely factored.

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

In Exercises \(133-136\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}}=49$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{2 x-1}{x-7}+\frac{3 x-1}{x-7}-\frac{5 x-2}{x-7}=0$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. In an inequality such as \(5 x+4<8 x-5,\) I can avoid division by a negative number depending on which side I collect the variable terms and on which side I collect the constant terms.

Perform the indicated operations. Simplify the result, if possible. $$\frac{a b}{a^{2}+a b+b^{2}}+\left(\frac{a c-a d-b c+b d}{a c-a d+b c-b d} \div \frac{a^{3}-b^{3}}{a^{3}+b^{3}}\right)$$

Perform the indicated operations. Simplify the result, if possible. $$\left(2-\frac{6}{x+1}\right)\left(1+\frac{3}{x-2}\right)$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

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.