Problem 12 Simplify. $$ -\left(-2 x^{2}... [FREE SOLUTION]

Chapter 10: Problem 12

Simplify. $$ -\left(-2 x^{2}-x+1\right) $$

Short Answer

Expert verified

The expression simplifies to $2x^2 + x - 1$.

Step by step solution

Identify External Negative Sign

The expression given is $-(-2x^2 - x + 1)$. Notice the negative sign outside the parentheses. This indicates that we need to change the sign of each term inside the parentheses.

Distribute the Negative Sign

Distribute the negative sign to each term within the parentheses. This means multiplying each term by -1: $-1 imes (-2x^2) = 2x^2$ $-1 imes (-x) = x$ $-1 imes 1 = -1$.

Write the Simplified Expression

Combine the terms after changing their signs. The expression simplifies to:\[2x^2 + x - 1\].

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Key Concepts

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

Negative Sign Distribution

Distributing a negative sign is a common task in algebra, essential for simplifying expressions. When you have a negative sign in front of parentheses, as in $-(-2x^2 - x + 1)$, it implies that all terms within need their signs flipped. This is because multiplying each term by -1 affects the term's coefficient and flips its sign.

To do this:

Multiply $-2x^2$ by -1, resulting in $2x^2$.
Multiply $-x$ by -1, resulting in $x$.
Multiply $+1$ by -1, which becomes $-1$.

Therefore, each term's sign within the parenthesis is reversed, leading to the final output.
This operation sets the foundation for further simplification of polynomials or any expressions involving terms grouped in parentheses.

Polynomial Simplification

Simplifying polynomials involves combining like terms to form a more manageable expression. Once the negative sign has been distributed as previously discussed, the next step involves ensuring there are no further simplifications possible.

In this exercise, after distributing the negative sign, you end up with:

$2x^2$ from $-(-2x^2)$
$x$ from $-(-x)$
$-1$ from $-(1)$

These terms do not share any common factors or like terms and thus cannot be combined further. The expression $2x^2 + x - 1$ is considered fully simplified. In simplifying polynomials, always check for possible like terms which can be added or subtracted.

Mathematical Operations in Simplification

Performing mathematical operations correctly is crucial when simplifying expressions. In our example, this mainly involves applying multiplication and addition/subtraction.

Multiplication: When we multiply each term inside the parentheses by -1, this step changes their signs. It's the application of the basic property of numbers where a negative times a negative equals a positive, and a negative times a positive equals a negative.
Addition and Subtraction: After distributing the negative sign, check if there are terms you can add or subtract. Since there are no like terms here, the expression stays $2x^2 + x - 1$.

Using these operations carefully ensures the expression is simplified correctly. Make sure every term is adjusted appropriately during each operation to avoid mistakes. Simplification is often just a precise rearrangement of terms based on the basic operations of mathematics.

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91影视

Simplify. $$ -\left(-2 x^{2}-x+1\right) $$

Short Answer

Step by step solution

Identify External Negative Sign

Distribute the Negative Sign

Write the Simplified Expression

Key Concepts

Negative Sign Distribution

Polynomial Simplification

Mathematical Operations in Simplification

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