Problem 4 Add the following polynomials. ... [FREE SOLUTION]

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Prealgebra

Charles P. McKeague

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 9: Problem 4

Add the following polynomials. $$\left(5 a^{2}-2 a+1\right)+\left(3 a^{2}-4 a+2\right)$$

Short Answer

Expert verified

The sum of the polynomials is $8a^2 - 6a + 3$.

Step by step solution

Distribute any '+' Signs

In this problem, since we are directly adding, we don't need to distribute any negative signs, simply remove the parentheses: $(5a^2 - 2a + 1) + (3a^2 - 4a + 2)$.

Combine Like Terms

Group and add the coefficients of like terms together. We have $5a^2$ and $3a^2$ for the $a^2$ terms, $-2a$ and $-4a$ for the $a$ terms, and constant terms $1$ and $2$: - For $a^2$ terms: $5a^2 + 3a^2 = 8a^2$- For $a$ terms: $-2a - 4a = -6a$- For constant terms: $1 + 2 = 3$.

Write the Resulting Polynomial

After adding the like terms, combine them to write the resulting polynomial: $8a^2 - 6a + 3$.

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Key Concepts

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

Understanding Like Terms

When working with polynomials, you'll frequently encounter terms like $5a^2$ or $-2a$. These are called "like terms" when they share the same variable and degree. In our example:

$5a^2$ and $3a^2$ are like terms because they both have $a^2$.
Similarly, $-2a$ and $-4a$ are considered like terms since both include $a$.
Lastly, the constant terms $1$ and $2$ are also like terms as they have no variable associated with them.

This concept is crucial when adding or subtracting polynomials because you can only combine like terms. This means you add or subtract the coefficients of these terms while keeping the variable and its exponent unchanged. Recognizing and properly identifying like terms will streamline your polynomial operations.
It's all about matching the structure!

Unpacking Algebraic Expressions

In algebra, an expression is like a mathematical phrase that can include numbers, variables, and operators. Algebraic expressions form the foundation of operations, such as polynomial addition. In the case of our given problem, you are presented with two expressions within parentheses that we need to work with:
$(5a^2 - 2a + 1)$ and $(3a^2 - 4a + 2)$.
Here's what each part signifies:

**Terms**: Every distinct part of the expression separated by a '+' or '-' sign, such as $5a^2$.
**Coefficients**: The numbers in front of the variables, for instance, the '5' in $5a^2$.
**Constants**: Numbers on their own without variables, like '1' and '2' in our expressions.

By understanding each component of an algebraic expression, you'll be better equipped to solve problems across various types of mathematical operations. It's all about breaking down the components!

Combining Coefficients

One essential skill in manipulating polynomials is the ability to combine coefficients. Coefficients, the numerical part of each term, dictate the magnitude of the variables we are dealing with. In polynomial addition, combining coefficients involves:

First, identify the like terms. For example, $5a^2$ and $3a^2$ are like terms, sharing the variable $a^2$.
Next, simply add their coefficients: For the $a^2$ terms, add 5 and 3 to get 8, resulting in $8a^2$.
Do the same for the rest: $-2a - 4a$ becomes $-6a$, and the constants $1 + 2$ yield 3.

This process involves only basic arithmetic but requires careful attention to ensure terms are correctly aligned by type. By focusing on the coefficients and ensuring that they are properly added along with their respective variables, the resulting polynomial expression is both accurate and simplified鈥攋ust what you need to solve these expressions successfully!

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91影视

Add the following polynomials. $$\left(5 a^{2}-2 a+1\right)+\left(3 a^{2}-4 a+2\right)$$

Short Answer

Step by step solution

Distribute any '+' Signs

Combine Like Terms

Write the Resulting Polynomial

Key Concepts

Understanding Like Terms

Unpacking Algebraic Expressions

Combining Coefficients

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