/*! 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 67 Add $$ \begin{array}{r}{-12 ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 67

Add $$ \begin{array}{r}{-12 p^{2}+4 p-1} \\ {3 p^{2}+7 p-8} \\ \hline\end{array} $$

Short Answer

Expert verified
-9p^2 + 11p - 9

Step by step solution

01

Align the Polynomials Vertically

Write the given polynomials in a column, aligning like terms (terms with the same power of p) directly on top of each other.
02

Add the Coefficients of Each Like Term

Add the coefficients of the terms that are aligned. For the \(p^2\) terms: \(-12 + 3\). For the \(p\) terms: \(+4 + 7\). For the constant terms: \(-1 - 8\).
03

Write the Result

Combine the results from Step 2 to form the new polynomial. Sum of \(p^2\) terms: \(-12 + 3 = -9p^2\). Sum of \(p\) terms: \(+4 + 7 = 11p\). Sum of constant terms: \(-1 - 8 = -9\).

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.

aligning polynomials
When adding polynomials, it is crucial to align them correctly. This means placing polynomials in columns so that the terms with the same power of the variable (in this case, 'p') are directly on top of each other. This alignment helps to ensure you can add the coefficients of like terms accurately.

For instance, let's take the two polynomials from the exercise:
-12p虏 + 4p - 1 and 3p虏 + 7p - 8.

To align these polynomials properly:
  • Write the terms of the first polynomial on the top row: -12p虏, +4p, -1.
  • Write the terms of the second polynomial in the same column, ensuring that the powers of p are aligned: +3p虏, +7p, -8.

It would look something like this:
$$\begin{array}{r}{-12p^{2} + 4p - 1} \ {+3p^{2} + 7p - 8} \ \ \ \hline\end{array}$$

With this layout, you are ready to add the coefficients of each corresponding term.
adding coefficients
Once the terms are aligned, the next step is to add the coefficients of like terms.

Let's break down this step:
  • First, identify the coefficients of the {p虏} terms: -12 and +3. Add these together: -12 + 3 = -9.
  • Next, look at the {p} terms: +4 and +7. Add these: +4 + 7 = +11.
  • Finally, address the constant terms: -1 and -8. Here is the sum: -1 - 8 = -9.

By summing the coefficients of the like terms, you make sure each power of the variable p is accurately combined. Remember, positive and negative coefficients need careful attention; simple errors can be easily made.
combining like terms
Combining like terms involves rewriting the polynomial by summing the coefficients that are already added.

In the context of the exercise, after adding our coefficients, we found:
  • {p虏} terms sum to -9p虏.
  • {p} terms sum to +11p.
  • Constant terms sum to -9.

Final combined polynomial: -9p虏 + 11p - 9.

Each term has been simplified and combined with others of the same power, ensuring the polynomial is fully simplified.

To summarize:
  • Align your polynomials.
  • Add coefficients of like terms correctly.
  • Rewrite the polynomial to reflect the combined terms.


Understanding these steps will make polynomial addition a more straightforward and error-free process.

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 Exercises 65鈥�74, the factors involve fractions or decimals. Apply the methods of this sec- tion, and find each product. $$ \left(3 t-\frac{5}{4}\right)\left(3 t+\frac{5}{4}\right) $$

Find each product. $$ (3 r+2 s)\left(r^{3}+2 r^{2} s-r s^{2}+2 s^{3}\right) $$

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ \left(\frac{5 z^{3}}{2 a^{2}}\right)^{-3}\left(\frac{8 a^{-1}}{15 z^{-2}}\right)^{-3} $$

Solve each problem. In \(2009,\) the national debt of the U.S. government was about \(\$ 11.5\) trillion. Using 307.0 million as the population for that year, about how much was this per American? Write the amount in standard notation to the nearest dollar.

Write each number in standard notation. $$ 5.42 \times 10^{-4} $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

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.