/*! 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 38 Perform the indicated operations... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 38

Perform the indicated operations and simplify. $$ (1+2 x)\left(x^{2}-3 x+1\right) $$

Short Answer

Expert verified
The simplified expression is \(2x^3 - 5x^2 - x + 1\).

Step by step solution

01

Distribute the First Term

First, we need to distribute the '1' from the first bracket into the second bracket. This means multiplying each term of the polynomial \(x^2 - 3x + 1\) by 1: \[1(x^2 - 3x + 1) = x^2 - 3x + 1.\]
02

Distribute the Second Term

Next, distribute '2x' from the first bracket into the second bracket. Multiply each term of \(x^2 - 3x + 1\) by \(2x\):\[2x(x^2 - 3x + 1) = 2x^3 - 6x^2 + 2x.\]
03

Combine Like Terms

Now, add up all the terms obtained from the distributions:\[x^2 - 3x + 1 + 2x^3 - 6x^2 + 2x.\] Group them by similar degree: \[2x^3 + (x^2 - 6x^2) + (-3x + 2x) + 1.\]
04

Simplify the Expression

Combine the like terms in each group:- Combine \(x^2 - 6x^2\) to get \(-5x^2\).- Combine \(-3x + 2x\) to get \(-x\).The simplified expression is:\[2x^3 - 5x^2 - x + 1.\]

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.

Distributive Property
The distributive property is a crucial concept in polynomial multiplication. It involves distributing or multiplying each term in a given expression with every term in another expression. To apply the distributive property in this exercise, we first took each term from the expression \(1 + 2x\) and multiplied it with each term in the polynomial\(x^2 - 3x + 1\). This means:
  • 1 multiplied by each term in the second polynomial gives: \(1 \cdot (x^2 - 3x + 1) = x^2 - 3x + 1\)
  • 2x multiplied by each term in the second polynomial gives: \(2x \cdot (x^2 - 3x + 1) = 2x^3 - 6x^2 + 2x\)
Each part of the distribution acts as a mini multiplication series, ensuring all combinations of terms are considered. The goal is to ensure no term is left without being paired with another.
Combining Like Terms
Once each term from \(1+2x\) has been distributed across the terms of \(x^2 - 3x + 1\), the next task revolves around combining like terms. Like terms are terms that share the same variable raised to the same power. They can be added together.
  • Start by grouping like terms from your expression: \(x^2 + 2x^3 - 6x^2 - 3x + 2x + 1\)
  • Terms \(x^2\) and \(-6x^2\) are combined as \(-5x^2\)
  • Terms \(-3x\) and \(2x\) are combined as \(-x\)
Combining like terms involves simplifying the expression by adding or subtracting coefficients, not altering the variable expressions themselves. It simplifies the expression while maintaining its overall value.
Algebraic Simplification
After applying the distributive property and combining like terms, the final step requires algebraic simplification. This involves organizing the expression into its simplest form, making it easier to understand and work with.
  • After the combining process, the terms are rearranged according to their power: \((2x^3 - 5x^2 - x + 1)\)
  • Simplification shows each term clearly and in order of decreasing powers of \(x\)
This step is essential for presenting the expression in a conventional format that informs future operations or evaluations. Simplicity in algebra indicates clarity and efficiency in problem-solving and communication.

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

Is This Rationalization? In the expression 2\(/ \sqrt{x}\) we would eliminate the radical if we were to square both numerator and denominator. Is this the same thing as rationalizing the denominator?

Write each number in decimal notation. $$ 7.1 \times 10^{14} $$

Simplify the expression and eliminate any negative exponent(s). $$ \frac{x^{9}(2 x)^{4}}{x^{3}} $$

\(89-96\) m State whether the given equation is true for all values of the variables. (Disregard any value that makes a denominator zero.) $$ 2\left(\frac{a}{b}\right)=\frac{2 a}{2 b} $$

Let \(a, b,\) and \(c\) be real numbers with \(a > 0, b < 0,\) and \( c < 0 .\) Determine the sign of each expression. \(\begin{array}{ll}{\text { (a) } b^{5}} & {\text { (b) } b^{10} \quad \text { (c) } a b^{2} c^{3}} \\ {\text { (d) }(b-a)^{3}} & {\text { (e) }(b-a)^{4}} \quad {\text { (f) } \frac{a^{3} c^{3}}{b^{6} c^{6}}}\end{array}\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Mechanics 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.