/*! 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 11 \(\left(6 x^{2}+3\right)(12 x-4)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 11

\(\left(6 x^{2}+3\right)(12 x-4)\)

Short Answer

Expert verified
The expanded form of the given binomial expression is: \(72 x^{3} - 24 x^{2} + 36 x - 12\).

Step by step solution

01

Distribute the First Term of First Binomial

First, distribute the first term of the first binomial, \(6 x^{2}\), to both terms in the second binomial. This gives \(6 x^{2} \cdot 12 x\) and \(6 x^{2} \cdot -4\). When you simplify these, they become \(72 x^{3}\) and \(-24 x^{2}\).
02

Distribute the Second Term of First Binomial

Next, distribute the second term of the first binomial, \(+3\), to both terms in the second binomial. This gives \(3 \cdot 12 x\) and \(3 \cdot -4\). Simplifying these provides \(36 x\) and \(-12\).
03

Combine Like Terms

Lastly, combine each of the terms that were found. Putting them all together results in the final expanded form: \(72 x^{3} - 24 x^{2} + 36 x - 12\).

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 that consists of exactly two terms. In our exercise, we have two binomials: \(6x^2 + 3\) and \(12x - 4\). Each binomial is made up of its respective terms which are joined by either a plus or minus sign.
  • The first binomial, \(6x^2 + 3\), contains the term \(6x^2\), which is a squared term, and \(+3\), a constant.
  • The second binomial, \(12x - 4\), consists of \(12x\), a linear term, and \(-4\), another constant.
Recognizing binomials is crucial because it sets the stage for polynomial multiplication. When multiplying two binomials, every term in the first binomial needs to be multiplied by every term in the second binomial. This way of thinking leads us to the distribution method, which will be discussed next.
The Distribution Method
The distribution method, often called the distributive property, refers to the process of multiplying each term in one binomial by all the terms in the other binomial. This method ensures that each combination of terms is considered in the multiplication.First, you need to distribute the initial term of the first binomial through each term of the second binomial.
  • Using our example: Distribute \(6x^2\) through \(12x - 4\). This forms \(6x^2 \cdot 12x\) and \(6x^2 \cdot -4\).
After completing this, the same process is applied to the second term of the first binomial.
  • Distribute \(+3\) through \(12x - 4\). This results in \(3 \cdot 12x\) and \(3 \cdot -4\).
This method makes sure all components are paired and helps build up the final polynomial expression. Once this is complete, the task becomes one of organization, which leads us to combining like terms.
Combining Like Terms
Combining like terms is the final step in the polynomial multiplication process. This involves adding or subtracting terms that have the same variables raised to the same power. Doing this correctly simplifies the expression to its cleanest form.In our example, after distributing, the product looks like: \(72x^3\), \(-24x^2\), \(36x\), and \(-12\). These are four distinct terms that cannot be combined further because each has different powers or variables.If there had been terms with the same variable and exponent, you would:
  • Add or subtract their coefficients.
  • Keep the same variable and power in the term.
Combining like terms helps in expressing the polynomial in the simplest form, aiding in interpretation and further algebraic manipulations.

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

An equation of the line normal to the graph of \(y=\sqrt{\left(3 x^{2}+2 x\right)}\) at \((2,4)\) is (A) \(4 x+7 y=20\) (B) \(-7 x+4 y=2\) (C) \(7 x+4 y=30\) (D) \(4 x+7 y=36\)

\(\int x \sqrt{x+3} d x=\) (A) \(\frac{2(x+3)^{\frac{3}{2}}}{3}+C\) (B) \(\frac{2}{5}(x+3)^{\frac{5}{2}}-2(x+3)^{\frac{3}{2}}+C\) (C) \(\frac{3(x+3)^{\frac{3}{2}}}{2}+C\) (D) \(\frac{4 x^{2}(x+3)^{\frac{3}{2}}}{3}+C\)

Now evaluate the following integrals. \(\int\left(1+\cos ^{2} x \sec x\right) d x\)

Sketch the slope field for \(\frac{d y}{d x}=2 x\)

If \(\frac{d y}{d x}=\frac{\left(3 x^{2}+2\right)}{y}\) and \(y=4\) when \(x=2,\) then when \(x=3, y=\) (A) 18 (B) 58 (C) \(\pm \sqrt{74}\) (D) \(\pm \sqrt{58}\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Statistics

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.