/*! 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 42 Add or subtract as indicated. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 42

Add or subtract as indicated. $$ \left(5 y^{2}-3 y-1\right)-\left(2 y^{2}+y+1\right) $$

Short Answer

Expert verified
\(3y^{2} - 4y - 2\)

Step by step solution

01

Identify the Expression

We are given the problem: \( \left(5y^{2} - 3y - 1\right) - \left(2y^{2} + y + 1\right) \). Notice that we are subtracting two polynomials.
02

Distribute the Negative Sign

Apply the negative sign from the subtraction to each term inside the second parenthesis. This means changing the signs: \( -(2y^{2} + y + 1) = -2y^{2} - y - 1 \).
03

Combine Like Terms

Rewrite the expression without parentheses: \( 5y^{2} - 3y - 1 - 2y^{2} - y - 1 \). Now combine like terms:- Combine \(5y^{2}\) and \(-2y^{2}\): \(3y^{2}\)- Combine \(-3y\) and \(-y\): \(-4y\)- Combine \(-1\) and \(-1\): \(-2\).
04

Write the Simplified Expression

The simplified result of adding or subtracting the given polynomials is \(3y^{2} - 4y - 2\).

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.

Distributing Negative Sign
When subtracting polynomials, distributing the negative sign is a crucial initial step. It ensures each term within a polynomial changes its sign. Subtraction can be tricky because of how signs affect each term.Here's a quick guide:
  • Identify the polynomial being subtracted. It's enclosed in parentheses and follows the subtraction sign.
  • Change the sign of each term within this polynomial. If a term is positive, it becomes negative, and vice versa.
So, if you have \(\left(5y^{2} - 3y - 1\right) - \left(2y^{2} + y + 1\right)\), the negative sign in front of \(2y^{2} + y + 1\) distributes to give \(-2y^{2} - y - 1\). This distribution is important because it keeps the equation balanced as you move to the next steps.
Combining Like Terms
After distributing the negative sign, combining like terms is the next important step. "Like terms" are terms that have the same variables raised to the same power. These can be combined by simply adding or subtracting their coefficients.For example:
  • With terms \(5y^{2} - 2y^{2}\), the like terms are \(y^{2}\). Their coefficients are 5 and -2, thus, combine to \(3y^{2}\).
  • With \(-3y\) and \(-y\), both share the variable \(y\). Combine their coefficients to get \(-4y\).
  • For constant terms \(-1\) and \(-1\), simply add them together to get \(-2\).
By combining these like terms, the expression transforms from a longer list into a simplified, easily readable form with each distinct term type.
Simplifying Polynomials
Once all like terms are combined, the polynomial is simplified. This means presenting the polynomial in its tidiest form, with terms written in descending order of their degree.Let's consider our example:
  • You start with the expression: \(5y^{2} - 3y - 1 - 2y^{2} - y - 1\).
  • After combining like terms, you should have \(3y^{2} - 4y - 2\).
This is now the simplest form because:
  • "Like terms" have been combined.
  • The degrees of terms are in descending order: \(y^{2}\), \(y\), constant.
  • There are no more operations or reductions possible.
Simplifying polynomials this way makes them easier to understand and work with, especially in further mathematical operations or equations.

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

Subtract using a vertical format. $$ \begin{array}{r} 5 x^{3}-4 x^{2}+6 x-2 \\ -\left(3 x^{3}-2 x^{2}-x-4\right) \\ \hline \end{array} $$

Simplify each expression by combining any like terms. $$ y-10+y $$

Add or subtract as indicated. $$ (7 y+7)-(y-6) $$

Simplify each expression by performing the indicated operation. Explain how you arrived at each answer. a. \(2 y+y\) b. \(2 y \cdot y\) c. \(-2 y-y\) d. \((-2 y)(-y)\)

Fill in the boxes so that each statement is true. (More than one answer may be possible for these exercises.) $$ \left(x^{\square}\right)^{\square}=x^{-15} $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Decision 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.