/*! 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 Multiply. $$3 x\left(2 x^{2}-3... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 13

Multiply. $$3 x\left(2 x^{2}-3 x+4\right)$$

Short Answer

Expert verified
\(6x^3 - 9x^2 + 12x\)

Step by step solution

01

Distribute the First Term

The expression given is \(3x(2x^2 - 3x + 4)\). We need to distribute \(3x\) across each term inside the parentheses. Start with distributing \(3x\) to \(2x^2\):\[3x imes 2x^2 = 6x^3\].
02

Distribute Second Term

Next, distribute \(3x\) to the second term \(-3x\):\[3x imes -3x = -9x^2\].
03

Distribute Final Term

Finally, distribute \(3x\) to the constant term \(+4\):\[3x imes 4 = 12x\].
04

Combine All Parts

Add all the distributed parts together to form the final expression. Combine \(6x^3\), \(-9x^2\), and \(+12x\) to get:\[6x^3 - 9x^2 + 12x\].

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.

Distribution in Polynomial Multiplication
When multiplying polynomials, distribution is a key technique. It involves multiplying each term inside a bracket by a term outside. This helps us break down complex expressions. In the exercise, the term outside the parenthesis is \(3x\), and the terms inside are \(2x^2 - 3x + 4\).
To distribute, multiply \(3x\) with each individual term inside the brackets. This step is done one term at a time.
  • Multiply \(3x\) and \(2x^2\) to get \(6x^3\).
  • Multiply \(3x\) and \(-3x\) to get \(-9x^2\).
  • Multiply \(3x\) and \(4\) to get \(12x\).
This method breaks down the multiplication into manageable steps, making it easier to handle the polynomial.
Understanding Monomials
A monomial is a single term algebraic expression consisting of a constant, a variable, or a product of constants and variables raised to a power. In this exercise, \(3x\), \(2x^2\), and \(-3x\) are examples of monomials.
Each monomial encapsulates several parts:
  • A coefficient, which is the numerical part (e.g., \(3\) in \(3x\)).
  • A variable part, which consists of the variable (e.g., \(x\)) and its exponent.
Monomials are the building blocks of polynomials. When multiplying monomials, you multiply their coefficients and add their exponents if the variables are the same.
Algebraic Expressions
An algebraic expression is a collection of numbers, variables, and arithmetic operations. It does not have an equality sign like equations do. The given exercise features an algebraic expression that is a polynomial due to its multiple terms.
Here鈥檚 what you鈥檒l find in an expression like \(3x(2x^2 - 3x + 4)\):
  • Multiple terms (e.g., \(2x^2\), \(-3x\), and \(4\)).
  • Operations such as addition and multiplication.
Such expressions allow flexibility in mathematical problems, enabling us to model situations and solve problems by simplifying the expression using techniques like distribution.
Working with Exponents
Exponents denote repeated multiplication of a base number. In the expression \(3x^2\), \(2\) is the exponent of the variable \(x\).
Whenever you multiply terms with the same base, you add their exponents. For example:
  • In \(3x \times 2x^2\), both have the base \(x\), so add the exponents: \(1 + 2 = 3\), resulting in \(6x^3\).
Remember, each variable in a monomial carries its own exponent, even if it is not explicitly written; for instance, \(x\) is actually \(x^1\).
Understanding how to manipulate exponents is crucial for solving algebraic expressions efficiently.

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

Evaluate each expression using exponential rules. Write each result in standard notation. See Example 7. $$ \frac{8 \times 10^{-1}}{16 \times 10^{5}} $$

Simplify. $$ \left(2 a^{3}\right)^{3} a^{-3}+a^{11} a^{-5} $$

Explain how to check a polynomial long division result when the remainder is not \(0 .\)

Write each number in standard notation. See Example 6 The population of the world is \(6.067 \times 10^{9} .\) Write this number in standard notation. (Source: U.S. Bureau of the Census)

Multiply. See Section 5.1 $$ -z^{2} y(11 z y) $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

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.