/*! 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 23 In Exercises 15鈥�58, find each ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 23

In Exercises 15鈥�58, find each product. $$ (3 x+5)(2 x+1) $$

Short Answer

Expert verified
The product of the two binomials \( (3x + 5)(2x + 1) \) is \( 6x^2 + 13x + 5 \)

Step by step solution

01

Distribution

Distribute each term of the first binomial, \(3x+5\), to every term of the second binomial, \(2x+1\). This results in four terms: \( (3x \cdot 2x) + (3x \cdot 1) + (5 \cdot 2x) + (5 \cdot 1) \)
02

Simplification of Terms

Simplify each term from the previous step. This results in: \( 6x^2 + 3x + 10x + 5 \)
03

Combination of Like Terms

Combine like terms (terms that have the same variable with the same exponent) to simplify the expression. In this case, combine the terms \(3x\) and \(10x\), which gives the final expression: \( 6x^2 + 13x + 5 \)

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.

Understanding Binomials
A binomial is a polynomial with exactly two terms. In the exercise, the expression is \(3x + 5\), which is a binomial because it has two distinct terms combined by a plus sign. Binomials can include numbers and variables, like \(3x\) and \(5\), with numbers representing coefficients and variables representing unknown values.
  • The first term \(3x\) is called a 'linear term,' since it involves a variable raised to the first power.
  • The second term \(5\) is a 'constant term,' since it doesn't include a variable.
When multiplying two binomials, such as \((3x + 5)\) and \((2x + 1)\), each term in one binomial must be multiplied by each term in the other. This is done through a procedure called distribution.
The Distribution Process
Distribution is a technique used to multiply each term in one binomial by each term in another to create a full expression. Imagine spreading the terms across each other, which is why it's called distribution. To multiply two binomials like \((3x + 5)(2x + 1)\), you follow these steps:
  • Multiply the first term in the first binomial, \(3x\), by each term in the second binomial, \(2x\) and \(1\).
  • Then multiply the second term of the first binomial, \(5\), by each term in the second binomial, \(2x\) and \(1\).
This results in four products, also known as 'partial products': - \(3x \cdot 2x = 6x^2\)- \(3x \cdot 1 = 3x\)- \(5 \cdot 2x = 10x\)- \(5 \cdot 1 = 5\)
Distribution allows us to methodically and systematically expand the product of two binomials into four separate terms.
Combining Like Terms Simplified
After distribution, you often end up with several terms in your polynomial expression. Many of these terms might be similar, known as 'like terms.' Like terms have the same variable with the same exponent.
For instance, after distributing \((3x + 5)(2x + 1)\), you get the expression \(6x^2 + 3x + 10x + 5\). To simplify, we need to combine the like terms, which here are the 'linear terms' \(3x\) and \(10x\).
  • Add the coefficients of the like terms together while maintaining the variable, which simplifies the middle terms to \(13x\).

  • The simplified polynomial expression is then \(6x^2 + 13x + 5\).

    • Combining like terms helps in making the polynomial much simpler and more manageable.

    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

    Suppose that a polynomial contains four terms. Explain how to use factoring by grouping to factor the polynomial.

    Simplify by reducing the index of the radical. $$ \sqrt[9]{x^{6}} $$

    If you are given two real numbers, explain how to determine which is the lesser.

    Will help you prepare for the material covered in the next section. A. Use a calculator to approximate \(\sqrt{300}\) to two decimal places. B. Use a calculator to approximate \(10 \sqrt{3}\) to two decimal places. C. Based on your answers to parts (a) and (b), what can you conclude?

    Determine whether each statement is trueor false. If the statement is false, make the necessary change(s) toproduce a true statement. $$x^{3}-64=(x+4)\left(x^{2}+4 x-16\right)$$
    See all solutions

    Recommended explanations on Math Textbooks

    Pure Maths

    Read Explanation

    Theoretical and Mathematical Physics

    Read Explanation

    Applied Mathematics

    Read Explanation

    Logic and Functions

    Read Explanation

    Discrete 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.