Problem 7 Add. $$ (x 2-3 x+7)+(3 x 2-8... [FREE SOLUTION]

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Elementary Algebra

John Redden

$Math Studyset 91影视 Explanations$ Math

2011 Edition

Chapter 5: Problem 7

Add. $$ (x 2-3 x+7)+(3 x 2-8 x-5) $$

Short Answer

Expert verified

The sum is $4x^2 - 11x + 2$.

Step by step solution

Identify like terms

The given expression is $(x^2 - 3x + 7) + (3x^2 - 8x - 5)$. Identify the like terms; these are terms with the same variables raised to the same powers. Here, like terms are those containing $x^2$, those containing $x$, and constant terms.

Combine like terms

Add the coefficients of the like terms. For $x^2$, add 1 and 3 to get 4. For $x$, add -3 and -8 to get -11. For the constant terms, add 7 and -5 to get 2.

Write the simplified expression

After combining the like terms, write the new expression: \[4x^2 - 11x + 2\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Adding Polynomials

When adding polynomials, we essentially look at combining together multiple algebraic expressions. Each polynomial could contain variables and coefficients. To add them, it's important to focus on their structure and components first, paying special attention to the terms and their respective powers.

To begin with, organize the polynomials by writing them alongside each other, as shown in the given exercise:

First, notice that each polynomial in the expression is placed within parentheses. These parentheses indicate separate groups of terms initially, but they don't affect the addition process.
Next, align any like terms so it's easy to see which can be combined.

After setting up the polynomial equations, proceed to add like terms, carefully ensuring that only terms with identical variables and exponents are combined. This will lead to a simplified expression.

Like Terms

In algebra, like terms are terms that have the exact same variables raised to the same powers. For example, in the expression provided, terms containing $x^2$, $x$, and constant numbers (with no variables) are grouped separately.

Analyzing the given example, we recognize the following:

Terms with $x^2$ are $x^2$ and $3x^2$.
Terms with $x$ are $-3x$ and $-8x$.
Constant terms are $+7$ and $-5$.

The process of combining all terms leads to the sum where coefficients are summed up to form new terms for each group. This process of neat categorization is vital to ensure that every aspect of the equation is distinctly grasped for simplification.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (such as addition or subtraction). In the context of polynomials, these expressions often feature terms like $x$, $x^2$, and constants. They form the building blocks of more complex algebraic problems.

In the exercise example:

Each polynomial is an algebraic expression on its own. They consist of a string of terms like $x^2 - 3x + 7$ and $3x^2 - 8x - 5$.
By adding these expressions, you are creating a new single algebraic expression, simplifying it to a cleaner, easier-to-read form.

Understanding the nature of algebraic expressions allows one to appreciate their usage in forming equations, modeling real-world situations, and performing effectively in any algebraic transition.

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91影视

Add. $$ (x 2-3 x+7)+(3 x 2-8 x-5) $$

Short Answer

Step by step solution

Identify like terms

Combine like terms

Write the simplified expression

Key Concepts

Adding Polynomials

Like Terms

Algebraic Expressions

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