/*! 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 29 Perform the indicated operations... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 29

Perform the indicated operations and simplify. $$ (2 x+3)(3 x-2) $$

Short Answer

Expert verified
The simplified expression is: \(6x^2 + 5x - 6\).

Step by step solution

01

Identify the terms

The given expression consists of two terms: \((2x + 3)\) and \((3x - 2)\).
02

Apply the distributive property (FOIL method)

FOIL stands for First, Outer, Inner, Last. We will multiply the terms using this method: First: Multiply the first terms of each term: \((2x)(3x)\) Outer: Multiply the outer terms of each term: \((2x)(-2)\) Inner: Multiply the inner terms of each term: \((3)(3x)\) Last: Multiply the last terms of each term: \((3)(-2)\)
03

Calculate each product

Now, we will calculate each of the products from Step 2: First: \((2x)(3x) = 6x^2\) Outer: \((2x)(-2) = -4x\) Inner: \((3)(3x) = 9x\) Last: \((3)(-2) = -6\)
04

Combine the products and simplify

Combine the products found in Step 3 and simplify by combining like terms: \(6x^2 - 4x + 9x - 6\) Combine like terms (-4x and 9x): \(6x^2 + 5x - 6\) The resulting simplified expression is: \(6x^2 + 5x - 6\)

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
When multiplying polynomials, one of the key concepts involved is the distributive property. This property allows us to multiply each term in one polynomial by each term in another polynomial. It's fundamental in handling expressions like \((a + b)(c + d)\), because we handle each part of the expressions individually to ensure nothing is forgotten.

To break it down: if we have two binomials like \((2x + 3)\) and \((3x - 2)\), using the distributive property, we multiply:
  • Each term in the first binomial by each term in the second binomial.
  • This includes setting up the products in sequences found in methods like FOIL (which we'll cover next).
This approach ensures you account for every combination of terms, leading us gradually to a simplified expression.
FOIL Method
The FOIL method is a handy tool for multiplying two binomials. It's a mnemonic that stands for: First, Outer, Inner, Last. This method ensures that you combine all parts of the two expressions correctly.

Let's apply it to our example, \((2x + 3)(3x - 2)\):
  • First: Multiply the first terms: \(2x \times 3x = 6x^2\)
  • Outer: Multiply the outer terms: \(2x \times -2 = -4x\)
  • Inner: Multiply the inner terms: \(3 \times 3x = 9x\)
  • Last: Multiply the last terms: \(3 \times -2 = -6\)
By systematically following this method, we ensure that no term combinations are overlooked. It simplifies the process of polynomial multiplication, making it intuitive and methodical.
Combining Like Terms
After multiplying the terms using the distributive property or FOIL method, the next step is to simplify the expression by combining like terms.

In our expression \(6x^2 - 4x + 9x - 6\), there are some terms we can combine:
  • The terms \(-4x\) and \(9x\) are 'like terms' because they both contain the variable \(x\) raised to the first power. When combined, they give \(5x\).
Thus, the simplified expression becomes:
  • \(6x^2 + 5x - 6\)
This step is crucial as it reduces the complexity of the polynomial and provides the final simplified form, making it easier to analyze or use in further calculations.

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

Solve the equation. $$ x^{4}-5 x^{2}+6=0 $$

Solve the equation. $$ 2 m^{2}+5 m+3=0 $$

Solve the equation. $$ \frac{3}{x^{2}-1}+\frac{2 x}{x+1}=\frac{7}{3} $$

Suppose \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are real numbers other than 0 and \(a>b\). State whether the inequality is true or false. $$ \frac{1}{a}>\frac{1}{b} $$

The quantity demanded \(x\) (measured in units of a thousand) of a certain commodity when the unit price is set at \(\$ p\) is given by the equation $$ p=\sqrt{-x^{2}+100} $$ If the unit price is set at \(\$ 6\), what is the quantity demanded?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

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.