/*! 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 17 Multiply and combine like terms.... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 17

Multiply and combine like terms. a. \(x(x+2)(2 x-1)\) (a) b. \((0.1 x-2.1)(0.1 x+2.1)\) (a)

Short Answer

Expert verified
a. \(2x^3 + 3x^2 - 2x\); b. \(0.01x^2 - 4.41\).

Step by step solution

01

Distribute First Pair

For part (a), start with distributing the first binomial. Multiply \(x\) by each term in \((x + 2)\):\[x imes (x + 2) = x^2 + 2x.\]
02

Distribute to Second Binomial

Next, distribute the result \(x^2 + 2x\) to the second binomial \((2x - 1)\):\[x^2(2x - 1) + 2x(2x - 1).\]
03

Multiply Both Distributions

Calculate the multiplication:1. \(x^2 imes 2x = 2x^3\)2. \(x^2 imes (-1) = -x^2\)3. \(2x imes 2x = 4x^2\)4. \(2x imes (-1) = -2x\)So, the expression becomes:\[2x^3 - x^2 + 4x^2 - 2x.\]
04

Combine Like Terms

Combine the like terms:\[2x^3 + 3x^2 - 2x.\]
05

Recognize Pattern in (b)

For part (b), use the difference of squares formula. The expression is \((0.1x - 2.1)(0.1x + 2.1)\). Recognize this as \((a - b)(a + b) = a^2 - b^2\), where \(a = 0.1x\) and \(b = 2.1\).
06

Apply Difference of Squares Formula

Apply the difference of squares formula:\[(0.1x)^2 - (2.1)^2.\]This evaluates to:\[0.01x^2 - 4.41.\]

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.

Polynomial Multiplication
Polynomial multiplication is a foundational concept in algebra. It involves distributing each term from one polynomial across every term in another polynomial. This operation is essential for simplifying expressions and solving equations that involve polynomials.

Let's take part (a) from the exercise, where we need to multiply the expression \(x(x+2)(2x-1)\). The process begins by distributing. Think of distribution as a way to share each term from one polynomial with every term in the other polynomial.
  • First, we distribute \(x\) to \((x + 2)\) to get \(x^2 + 2x\).
  • Next, we take this result and distribute it to the next binomial \((2x - 1)\), but we do it in parts to keep things organized.
  • Multiply \(x^2\) with each term in \((2x - 1)\), then do the same with \(2x\).
This results in four terms: \(2x^3, -x^2, 4x^2,\) and \(-2x\). Remember, it's just about systematically multiplying across each term while paying attention to the signs.
Combining Like Terms
Once the polynomial multiplication is done, the next step is to simplify the expression by combining like terms. Like terms are terms that have exactly the same variable parts raised to the same power, but they can have different coefficients.

In our example, after multiplying, we end up with the expression \(2x^3 - x^2 + 4x^2 - 2x\). Here's how to combine them:
  • Identify terms with the same variable and exponent.
  • Add or subtract the coefficients of like terms.
  • In our example, \(-x^2\) and \(4x^2\) are like terms because they both have \(x^2\).
  • Combine them to get \(3x^2\).
So, the simplified expression becomes \(2x^3 + 3x^2 - 2x\). When combining like terms, remember to only focus on the coefficients and keep the variable part unchanged.
Difference of Squares
The difference of squares is a special pattern in algebra that helps simplify expressions quickly without performing lengthy calculations. It applies when you have two terms, one subtracted from the other, both squared. The formula is \((a - b)(a + b) = a^2 - b^2\).

In part (b) of our exercise, we apply this principle to \((0.1x - 2.1)(0.1x + 2.1)\). By recognizing it as a difference of squares, we can directly use the formula:
  • Set \(a = 0.1x\) and \(b = 2.1\).
  • Substitute into the formula to get \((0.1x)^2 - (2.1)^2\).
  • This simplifies to \(0.01x^2 - 4.41\).
Using the difference of squares not only saves time but also helps prevent mistakes in calculation, as it is a straightforward formula-based solution to what could otherwise be a more complex multiplication problem.

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

In \(10 a-c\), you are given the midpoint of a segment and one endpoint. Find the other endpoint. a. midpoint: \((7,4)\) endpoint: \((2,4)\) (a) b. midpoint: \((9,7)\) endpoint: \((15,9)\) c. midpoint: \((-1,-2)\) endpoint: \((3,-7.5)\)

Draw four congruent rectangles-that is, all the same size and shape. a. Shade half the area in each rectangle. Use a different way of dividing the rectangle each time. b. Which of your methods in 13 a divide the rectangle into congruent polygons? c. Ripley divided one of her rectangles like this. Is the area divided in half? Explain.

On graph paper or your calculator, draw a triangle with vertices \(A(11,6), B(4,-8)\), and \(C(-6,6)\). a. Find the midpoint of each side. Label the midpoint of \(\overline{A B}\) point \(D\), the midpoint of \(\overline{B C}\) point \(E\), and the midpoint of \(\overline{C A}\) point \(F\). b. Find the slope of the segment from each vertex to the midpoint of the side opposite that vertex. c. Write an equation for each median. (a) d. Solve a system of equations to find the intersection of median \(\overline{A E}\) and median \(\overline{B F}\). e. Solve a system of equations to find the intersection of median \(\overline{A E}\) and median \(\overline{C D}\). f. What conjecture can you write, based on your answers to \(9 \mathrm{~d}\) and \(\mathrm{e}\) ? g. Did you use inductive or deductive reasoning to write your conjecture in \(9 f\) ?

Strips of graph paper may help in \(10 \mathrm{a}\). a. Draw or make triangles with the side lengths that are given in i-iv. Then use a protractor or the corner of a sheet of paper to find whether each triangle is a right triangle. i. \(5,12,13\) (a) ii. \(7,24,25\) iii. \(8,10,12\) iv. \(6,8,10\) b. Based on your results from \(10 \mathrm{a}\), does the Pythagorean Theorem seem to work in reverse? In other words, if the relationship \(a^{2}+b^{2}=c^{2}\) is true, is the triangle necessarily a right triangle?

The population of City A is currently 47,000 and is increasing at a rate of \(4.5 \%\) per year. The population of City B is currently 56,000 and is decreasing at a rate of \(1.2 \%\) per year. a. What will the populations of the two cities be in 5 years? b. When will the population of City A first exceed 150,000 ? c. If the population decrease in City B began 10 years ago, how large was the population before the decline started?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

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.