Problem 68 Perform the indicated operations... [FREE SOLUTION]

Chapter 5: Problem 68

Perform the indicated operations. $$\left(w^{2}-3 w+2\right)+\left(2 w-3+w^{2}\right)$$

Short Answer

Expert verified

The simplified expression is $2w^2 - w - 1$.

Step by step solution

Distribute and Group Like Terms

First, rewrite the expression \[ \big(w^2 - 3w + 2\big) + \big(2w - 3 + w^2\big) \]by combining all the like terms together.

Combine Like Terms

Combine the terms with $w^2$, $w$, and the constants separately:\[ w^2 + w^2 - 3w + 2w + 2 - 3 \]

Simplify Each Group of Like Terms

First, add the $w^2$ terms:\[ w^2 + w^2 = 2w^2 \]Next, combine the $w$ terms:\[ -3w + 2w = -w \]Finally, combine the constant terms:\[ 2 - 3 = -1 \]

Write the Final Expression

Put all the simplified terms together to get the final expression:\[ 2w^2 - w - 1 \]

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Key Concepts

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

Combining Like Terms

Combining like terms is a fundamental process in algebra. When you have an expression with multiple terms, you can only combine those that have the same variable raised to the same power.
For example, in the expression $w^2 - 3w + 2 + 2w - 3 + w^2$, the like terms are $w^2$, $-3w$, $2w$, $2$, and $-3$.
To combine them:

Group the $w^2$ terms: $w^2 + w^2$
Group the $-3w$ and $2w$: $-3w + 2w$
Group the constants $2$ and $-3$

After grouping, you perform the arithmetic operations to each group: $w^2 + w^2 = 2w^2, -3w + 2w = -w,$ and $2 - 3 = -1$.

Simplifying Polynomials

Simplifying polynomials involves reducing your expression to its simplest form. After combining like terms, we need to add or subtract these terms.
In our example, we start with: $(w^2 -3w + 2) + (2w - 3 + w^2)$.
After combining like terms, we get: $(w^2 + w^2) + (-3w + 2w) + (2 - 3)$.
This simplifies to $2w^2 - w - 1$.
The final expression $2w^2 - w - 1$ is the simplest form of the polynomial after combining all like terms and carrying out the necessary arithmetic operations.

Algebraic Expressions

Algebraic expressions represent relationships between numbers and variables. They include numbers, variables (like $w$), and operations (like addition and subtraction).
For instance, in the expression $w^2 -3w + 2$, $w^2$ is a term with a variable raised to the second power, $-3w$ is a term with a variable raised to the first power, and $2$ is a constant term.
When working with algebraic expressions, it鈥檚 important to understand the properties of operations and the concept of like terms to simplify and solve problems efficiently. This understanding is key to mastering polynomial operations.

Addition of Polynomials

Adding polynomials is about combining like terms from each polynomial.
In our exercise, we add the polynomials $w^2 - 3w + 2$ and $2w - 3 + w^2$.
Steps to add polynomials:

Write down both polynomials.
Combine the like terms: $w^2 + w^2, -3w + 2w, 2 - 3$.
Simplify each group of like terms: $2w^2, -w, -1$.

The final result is the simplified polynomial: $2w^2 - w - 1$.
Understanding and practicing the addition of polynomials helps build a solid foundation for more complex algebraic operations.

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91影视

Perform the indicated operations. $$\left(w^{2}-3 w+2\right)+\left(2 w-3+w^{2}\right)$$

Short Answer

Step by step solution

Distribute and Group Like Terms

Combine Like Terms

Simplify Each Group of Like Terms

Write the Final Expression

Key Concepts

Combining Like Terms

Simplifying Polynomials

Algebraic Expressions

Addition of Polynomials

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