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Chapter 5: Problem 34

Simplify. $$ -\left(-6 a^{3}+2 a^{2}-7\right) $$

Short Answer

Expert verified
6a^3 - 2a^2 + 7

Step by step solution

01

Understand the negative sign outside the parentheses

There is a negative sign outside the parentheses that indicates we need to distribute it to each term inside the parentheses. The negative sign will change the sign of each term.
02

Distribute the negative sign

Distribute the negative sign to each term inside the parentheses: -(-6a^3 + 2a^2 - 7) = -(-6a^3) + -(2a^2) - (-7)
03

Change the signs of each term

Change the sign of each term within the parentheses: -(-6a^3) = 6a^3 -(2a^2) = -2a^2 -(-7) = 7
04

Combine the results

Combine the terms according to the changes made: 6a^3 - 2a^2 + 7

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Key Concepts

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

Negative Sign Distribution
Let's start by exploring how to deal with a negative sign outside parentheses, a crucial step in simplifying polynomials. When you see a negative sign before a set of parentheses, it signals that you must apply this sign to every term inside.

Think of it as multiplying each term by -1. For instance, in the expression \( -(-6a^3 + 2a^2 - 7) \),
you need to distribute the negative sign:

\[ -(-6a^3) - (2a^2) - (-7) \].
Notice how each term inside the parentheses changes its sign. This step is vital for accurate simplification.
Polynomial Terms
Now that we've applied the negative sign, let's talk about polynomial terms.
A polynomial consists of terms like
\( 6a^3, -2a^2, \) and \(7 \).

Each term has a coefficient (the number in front) and a variable part (like
\(a^3 \) or \(a^2\)).
When simplifying, we distribute the negative sign to affect both these parts. For example,
\( -(-6a^3) \); multiplying through makes it \( 6a^3 \).
Think of it as flipping signs. Positive becomes negative, and vice versa.
Simply keep track of these changes to handle polynomial terms properly.
Sign Change
Understanding sign changes is the last piece of the puzzle. Here's what you need to know:
  • Positive to Negative: If a term inside parentheses is positive, as in
    \(2a^2\)
    , distributing a negative sign makes it negative:
    \( -(2a^2) = -2a^2 \).
  • Negative to Positive: Alternatively, if a term is negative, like
    \(-7\), changing the sign makes it positive:
    \( -(-7) = 7 \). This is like subtracting a negative, which yields a positive.

By mastering sign changes, you'll simplify more accurately. Put all these steps together:
after applying the negative sign in \( -(-6a^3 + 2a^2 - 7)\),
you get \( 6a^3 - 2a^2 + 7 \).
By understanding and applying these core concepts, you're on your way to mastering polynomial simplifications!

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Most popular questions from this chapter

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Simplify. $$ 5^{0}-5^{-1} $$

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