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Chapter 5: Problem 69

Perform the indicated operations. $$\left(w^{2}-7 w-2\right)-\left(w-3 w^{2}+5\right)$$

Short Answer

Expert verified
4w^2 - 8w - 7

Step by step solution

01

Distribute the negative sign

First, distribute the negative sign to each term inside the second set of parentheses: - (w - 3w^2 + 5). This becomes -w + 3w^2 - 5.
02

Rewrite the expression

Now, rewrite the original expression by combining the terms as: (w^2 - 7w - 2) - (w - 3w^2 + 5) becomes w^2 - 7w - 2 - w + 3w^2 - 5
03

Combine like terms

Now, combine the like terms: -2 + (-5) = -7, w^2 + 3w^2 = 4w^2, -7w - w = -8w. So the final expression is: 4w^2 - 8w - 7

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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 an essential part of simplifying polynomial expressions. Like terms are terms in a polynomial that have the same variable raised to the same power.

For example, in the expression 4w^2 - 8w - 7, we have:
  • 4w^2 and 3w^2 are like terms because they both contain w raised to the power of 2.
  • -7w and -w are like terms because they both contain w raised to the power of 1.
By adding or subtracting the coefficients (the numbers in front of the variables), you can combine these like terms to simplify the expression.
Distributive Property
The distributive property allows you to multiply a single term across multiple terms within parentheses. This is crucial when dealing with polynomial expressions. The property states that a(b + c) = ab + ac.

In our exercise, we use the distributive property in step 1:
  • We have -(w - 3w^2 + 5).
  • Distributing the negative sign, we get -w + 3w^2 - 5.
Distributing effectively 'applies' the negative sign (or any other multiplier) to each term inside the parentheses, making it easier to combine like terms later.
Simplifying Expressions
Simplifying expressions involves rewriting them in the simplest form possible. This often means combining like terms and applying the distributive property.

Let's break down the final simplification in our given problem:
  • After applying the distributive property, the expression became: w^2 - 7w - 2 - w + 3w^2 - 5.
  • We identified and combined the like terms properly:
    • w^2 and 3w^2 = 4w^2.
    • -7w and -w = -8w.
    • -2 and -5 = -7.
  • This results in the final simplified expression: 4w^2 - 8w - 7.
Simplifying helps make the expression more manageable and often reveals patterns or solutions more clearly.

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