/*! 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 13 In Exercises \(11-16,\) factor b... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 13

In Exercises \(11-16,\) factor by grouping. $$x^{3}-x^{2}+2 x-2$$

Short Answer

Expert verified
The factored form of the expression \(x^{3}-x^{2}+2 x-2\) by grouping is \((x - 1)(x^2 + 2)\).

Step by step solution

01

Grouping the terms

The given expression is \(x^{3}-x^{2}+2 x-2\). This can be grouped into two separate parts as follows \(x^{3}-x^{2}\) and \(+2 x-2\).
02

Factoring out the GCF from each group

From the first group \(x^{3}-x^{2}\), the GCF is \(x^{2}\), factoring out this, we get \(x^2(x -1)\). From the second group \(2x - 2\), the GCF is 2, factoring out this, we get \(2(x -1)\).
03

Factoring by grouping

Finally, observe that the terms we ended up after factoring the GCF from each group, \(x^2(x -1)\) and \(2(x -1)\), contain a common factor of \(x -1\). Factor out this common factor, you obtain the final factored form, \((x - 1)(x^2 + 2)\).

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)
When dealing with algebraic expressions, finding the Greatest Common Factor (GCF) is an essential step in simplifying and factoring. The GCF is the largest quantity that divides two or more terms without leaving a remainder. This concept is akin to finding the GCF in basic arithmetic but applied to algebraic terms.

**How to Find the GCF:**
  • Identify each term's prime factors, including variables with their lowest exponents.
  • For each group, pick the lowest power for each common variable or coefficient.
  • The GCF is the product of these selected factors.
In the expression from our example, for \(x^3 - x^2\), the GCF is \(x^2\) as it is the highest power of \(x\) common to both terms. Similarly, in \(2x - 2\), the number 2 is the GCF.
Polynomials
Polynomials are expressions involving variables raised to whole number powers and coefficients. They're a fundamental concept in algebra and can take many forms depending on the number of terms they contain.

**Components of Polynomials:**
  • **Monomial:** A single term (e.g., \(7x\)).
  • **Binomial:** Two terms (e.g., \(3x + 2\)).
  • **Trinomial:** Three terms (e.g., \(x^2 + 5x + 6\)).
Our primary example, \(x^3 - x^2 + 2x - 2\), is a polynomial with four terms. Each term has its own degree based on the exponent of the variable.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations (like addition, subtraction, etc.). They form the backbone of algebra and are used to describe relationships and solve problems.

**Understanding Algebraic Expressions:**
  • Composed of terms, which can include constants, variables, or both.
  • Operations such as addition, subtraction, multiplication, and division combine the terms.
  • Expressions can be simplified or factored to solve equations or find specific values.
In the exercise, we started with the expression \(x^3 - x^2 + 2x - 2\), showcasing the use of these basic operations and variables in forming complex expressions. Factoring by grouping was the method used to simplify and transform this expression into a product of simpler ones.

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 \(11-16,\) factor by grouping. $$x^{3}-2 x^{2}+5 x-10$$

Use the quotient rule to simplify the expressions in Exercises \(17-26 .\) Assume that \(x>0\) $$\frac{\sqrt{72 x^{3}}}{\sqrt{8 x}}$$

simplify each algebraic expression. $$ 4(5 y-3)-(6 y+3) $$

In Exercises \(1-10\), factor out the greatest common factor. $$x^{2}(2 x+5)+17(2 x+5)$$

Simplify each exponential expression $$ \frac{8 x^{20}}{2 x^{4}} $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

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.