/*! 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 39 Find each sum or difference. $... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 39

Find each sum or difference. $$\left(9 a+3 a^{2}\right)+(a+4)$$

Short Answer

Expert verified
The sum is \(3a^2 + 10a + 4\).

Step by step solution

01

Identify Like Terms

The expression given is \((9a + 3a^2) + (a + 4)\). Before proceeding with addition, identify the like terms: \(9a\) and \(a\) are like terms as both have the variable \(a\), and \(3a^2\) is a separate term as it is of a different degree. The constant \(4\) has no like terms in the expression.
02

Group Like Terms

Rewrite the expression by grouping like terms together: \((9a + a) + 3a^2 + 4\). This organizing makes it easier to combine terms that are similar.
03

Simplify Like Terms

Add the coefficients of the like terms: \(9a + a = 10a\). The expression now reads \(10a + 3a^2 + 4\).
04

Finalize the Expression

Since there are no more like terms to combine, the simplified and final expression is \(3a^2 + 10a + 4\).

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.

Like Terms
When working with polynomials, the concept of "like terms" is fundamental to simplifying expressions. Like terms are terms that have the same variable raised to the same power. This means that only the coefficients鈥攖he numerical part in front鈥攃an differ.
For example, in the polynomial
  • \(9a\) and \(a\)
  • \(3a^2\)
we can see that \(9a\) and \(a\) are like terms because both share the same variable \(a\) raised to the first power. However, \(3a^2\) is not a like term with \(9a\) or \(a\) because it has a different exponent (it is \(a^2\), not \(a\)).
Finding like terms is the first step in combining them to simplify an algebraic expression.
Simplifying Expressions
Simplifying an algebraic expression involves combining like terms and organizing it in a way that makes it easier to understand and work with.
By identifying like terms, we can add or subtract their coefficients to simplify the polynomial. Consider the expression:
  • Initial Expression: \((9a + a) + 3a^2 + 4\)
  • Simplified Like Terms: \(9a + a = 10a\)
  • Simplified Expression: \(10a + 3a^2 + 4\)
After combining the like terms, we've reduced the expression to its simplest form. Whenever simplifying expressions, remember:
  • Combine only like terms
  • Ensure each term follows the variable and power rules
  • Check your work by verifying each term is correctly simplified
Algebraic Expressions
An algebraic expression is a mathematical phrase that can include numbers, variables, and operations, but no equality sign. These expressions are foundational in algebra and are used to represent problems and their solutions:
  • Variable Terms: Can contain numbers and letters (like \(9a\) and \(3a^2\))
  • Constants: Are the numbers without variables (like \(4\))
In our exercise, the expression \((9a + 3a^2) + (a + 4)\) contains both variable terms and a constant. The goal with algebraic expressions often involves rearranging and simplifying them. This helps solve for unknowns or make them more digestible for further mathematical operations.
Being comfortable with these expressions requires practice in recognizing patterns, grouping terms, and performing arithmetic operations on the coefficients.

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

Rewrite each expression using parentheses so that the terms having variables of the same power are grouped together. $$3 x^{2}-1+x^{2}$$

Graph each pair of equations on the same coordinate plane. Describe their similarities and differences. $$\begin{aligned} &y=0.5 x^{3}\\\ &y=2 x^{3} \end{aligned}$$

Find each difference. $$(2 x+7)-(x-1)$$

Find each product. $$(a-2 b) a b$$

Identify the equation that doesn't belong with the other three. A. \(y=\frac{1}{3} x-x\) B. \(y^{2}=3 x\) C. \(5 x+y=6\) D. \(x=5 y\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

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.