/*! 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 40 Add. \(\begin{array}{r}{-2 x^{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 40

Add. \(\begin{array}{r}{-2 x^{2}+3 x-9} \\ {+\quad(2 x-3)} \\ \hline\end{array}\)

Short Answer

Expert verified
The result is \(-2x^2 + 5x - 12\).

Step by step solution

01

Align the Terms

Write the polynomials in a vertical format to align the like terms. We have:\[\begin{array}{r}-2x^{2} + 3x - 9\+(0x^2 + 2x - 3)\end{array}\]Note that the term \(0x^2\) is added to the second polynomial to align the terms correctly.
02

Add the Like Terms

Perform the addition by combining the coefficients of like terms (terms with the same degree):- **Constant term (-9 and -3):** \(-9 + (-3) = -12\)- **Linear term (3x and 2x):** \(3x + 2x = 5x\)- **Quadratic term (-2x^2 and 0x^2):** \(-2x^2 + 0x^2 = -2x^2\)So, the result is: \(-2x^2 + 5x - 12\).
03

Write the Final Expression

Combine the results from step 2 into a single polynomial expression:\[-2x^2 + 5x - 12\]

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 Like Terms
Like terms in a polynomial are terms that have the exact same variable raised to the same power. This means these terms can be directly combined by adding or subtracting their coefficients.
For instance, in the expression given, the terms \(3x\) and \(2x\) are like terms because they both have the variable \(x\) raised to the power of 1.
Similarly, \(-2x^2\) and \(0x^2\) are also like terms because they both involve \(x^2\).
Understanding like terms is crucial because it simplifies complex polynomials by reducing the number of terms, making the equation easier to handle.
When handling polynomials, always identify like terms to effectively combine them later on.
Arranging Polynomials in Vertical Format
Arranging polynomials vertically helps in aligning like terms for easy addition or subtraction.
This format ensures that every term of the polynomials is lined up according to its degree or the power of the variable.
For example, when we write:
  • \(-2x^2 + 3x - 9\)
  • \(0x^2 + 2x - 3\)
The terms are arranged directly above each other, with quadratic (\(x^2\)), linear (\(x\)), and constant terms lined up.
This simplification makes it easy to see which coefficients belong to like terms. Knowing this allows us to accurately perform operations on each column, reducing the chance of errors.
Combining Coefficients Correctly
When you add polynomials, combining coefficients is simply adding or subtracting the numbers in front of like terms.
Coefficients are the numerical parts of terms and tell us how much of each variable we have.
In the example:
  • The constant terms \(-9\) and \(-3\) add up to \(-12\).
  • The linear terms \(3x\) and \(2x\) combine to \(5x\).
  • The quadratic terms \(-2x^2\) and \(0x^2\) remain \(-2x^2\).
Correctly combining these coefficients gives the final summed polynomial expression, \(-2x^2 + 5x - 12\).
Remember, the key lies in focusing on the numerical coefficients and their respective signs to ensure accuracy in addition or subtraction.

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

If \(P(x)\) is the polynomial given, find a. \(P(a),\) b. \(P(-x),\) and c. \(P(x+h)\). \(P(x)=-4 x\)

Factor each polynomial completely. See Examples 1 through 12. $$ 2 x^{2}+16 x y+32 y^{2} $$

Recall that a graphing calculator may be used to check addition, subtraction, and multiplication of polynomials. In the same manner, a graphing calculator may be used to check factoring of polynomials in one variable. For example, to see that $$ 2 x^{3}-9 x^{2}-5 x=x(2 x+1)(x-5) $$ graph \(\mathrm{Y}_{1}=2 x^{3}-9 x^{2}-5 x\) and \(\mathrm{Y}_{2}=x(2 x+1)(x-5) .\) Then trace along both graphs to see that they coincide. Factor the following and use this method to check your results. $$ x^{3}+6 x^{2}+8 x $$

Factor each polynomial completely. See Examples 1 through 12. $$ 36 x y^{2}-48 x y z^{2}+16 x z^{4} $$

Factor each polynomial completely. See Examples 1 through 12. $$ 3 x^{2}-6 x+3 $$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Decision Maths

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.