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Chapter 5: Problem 4

Simplify: \(-\left(2 x^{2}-3 x+9\right)\)

Short Answer

Expert verified
The expression simplifies to \(-2x^2 + 3x - 9\).

Step by step solution

01

Distribute the Negative Sign

Start by applying the negative sign to each term inside the parentheses. This means you will multiply each term within the parentheses by -1. The expression inside is \(2x^2 - 3x + 9\). Thus, it becomes:\(-1(2x^2) + (-1)(-3x) + (-1)(9)\).
02

Simplify Each Term

Perform the multiplication for each term:1. \(-1 \cdot 2x^2 = -2x^2\)2. \(-1 \cdot (-3x) = 3x\)3. \(-1 \cdot 9 = -9\)
03

Combine Simplified Terms

Write down the simplified expression by arranging the terms obtained in the previous step:\(-2x^2 + 3x - 9\)

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Key Concepts

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

Distributive Property
The distributive property is a fundamental concept in algebra that helps in simplifying expressions. Essentially, it means multiplying a single term by each term inside a set of parentheses. This operation is essential because it allows us to eliminate the parentheses and expand expressions in a structured way.

Here's what you should remember:
  • The general application is expressed as: \(a(b + c) = ab + ac\).
  • It applies not only to addition inside the parentheses but also to subtraction: \(a(b - c) = ab - ac\).
Using the distributive property correctly leads to more manageable expressions, getting rid of complex nested parentheses setups. As you distribute, ensure that each part is addressed, and remember this works similarly when the initial term is negative.
Polynomial Simplification
Simplifying polynomials involves combining like terms and eliminating unnecessary complexity to make expressions easier to work with. It is a crucial skill in algebra since it paves the way for solving equations and performing calculus operations later on.

When working to simplify:
  • Look for like terms, which are terms that have the same variables raised to the same power. For example, in the expression \(3x^2 + 5x - 2x^2\), the terms \(3x^2\) and \(-2x^2\) are like terms.
  • Add or subtract the coefficients of these like terms to combine them.
  • Once like terms are combined, arrange the terms in a standard order, often starting with the highest power of x (e.g., quadratic term, linear term, constant).
Properly simplified expressions are not only easier to handle but also provide clearer insight into the solution.
Negative Sign Distribution
Distributing negative signs can be tricky but is crucial for accurately simplifying expressions. This process involves applying a negative sign to each term inside the parentheses to effectively "flip" each term's sign.

Here's what to keep in mind:
  • Consider the negative sign as multiplying each term in the parentheses by \(-1\).
  • Each positive term becomes negative, and each negative term becomes positive.
  • This is crucial when parentheses include subtraction or when subtractive parentheses are encountered in wider expressions.
In practice, managing negative signs carefully prevents errors and ensures accuracy in solutions. Thus, being meticulous with this part of simplification is always beneficial.

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Most popular questions from this chapter

Given the rule \(f: x \rightarrow 3 x-5\), answer the question "where does \(f\) send \(-2 ?^{\prime \prime}\)

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Find the zero(s) of the function \(f(x)=2.6 x-9.62\).

Simplify: \(\left(-5 a^{2} b+4 a b-3 a b^{2}\right)+\left(2 a^{2} b+7 a b-a b^{2}\right)\)

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