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Chapter 5: Problem 35

Simplify. $$ -\left(-4 x^{4}+6 x^{2}+\frac{3}{4} x-8\right) $$

Short Answer

Expert verified
\(4x^4 - 6x^2 - \frac{3}{4}x + 8\right)\)

Step by step solution

01

- Distribute the Negative Sign

Distribute the negative sign to each term inside the parentheses. Multiply each term by \(-1\right)\). The original expression is \(-\right) left(-4x^{4}+6x^{2}+\frac{3}{4} x - 8\right)\)
02

- Multiply Each Term

When distributing the negative sign, each term within the parentheses will change its sign:\right) \(-1 (-4x^4)\) becomes \(4x^4\), \right) -1 (6x^2)\( becomes \right) -6x^2\), \right) -1 \frac{3}{4}x\( becomes \right) \frac{3}{4}x\right)\), and \right) -1 (-8)\( becomes \right) 8\right)\).
03

- Combine the Simplified Terms

Combine the simplified terms to obtain the final result: \(4x^{4}-6x^{2}-\frac{3}{4} x + 8\right)\).

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Key Concepts

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

Distribute Negative Sign
When you see a negative sign in front of a set of parentheses, it's important to distribute the negative sign to each term inside the parentheses. This means you'll multiply each term by \(-1\). This step changes the sign of each term. For example, in the expression \(-(-4x^{4}+6x^{2}+\frac{3}{4} x - 8)\), distributing the negative sign will affect each term as follows:
  • \(-1(-4x^4)\) becomes \(+4x^4\)
  • \(-1(6x^2)\) becomes \(-6x^2\)
  • \(-1(\frac{3}{4}x)\) becomes \(-\frac{3}{4}x\)
  • \(-1(-8)\) becomes \(+8\)
Distributing the negative sign is crucial to ensuring all terms are correctly simplified. Without this step, the expression is not properly simplified.
Combine Like Terms
After distributing the negative sign, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In the simplified expression \(4x^{4}-6x^{2}-\frac{3}{4} x + 8\), you would look for terms with the same variable and exponent. Here are some tips to identify and combine like terms:
  • Terms like \(4x^4\) and \(-6x^2\) are already separated because they have different powers of \x\.
  • If there were multiple \(x^2\) terms, you would add or subtract their coefficients.
  • Make sure to include the sign (positive or negative) when combining terms.
Combining like terms simplifies the expression to a form that's easier to understand and work with.
Simplify Polynomial Expressions
Simplifying polynomial expressions involves a combination of distributing signs and combining like terms. This process makes complex expressions easier to understand and solve. Here鈥檚 a quick summary of how you can simplify any polynomial expression effectively:
  • Start by eliminating any parentheses by distributing any required signs.
  • Identify and combine like terms, ensuring you keep track of positive and negative signs.
  • Write down the simplified polynomial with terms written starting from the highest degree to the lowest.
Let鈥檚 use our previously simplified expression, \(4x^{4}-6x^{2}-\frac{3}{4} x + 8\), as an example. Notice that each term is written in descending order of the exponent. Simplifying polynomial expressions is a foundational skill in algebra and helps in solving more complex equations later on. Each simplification step brings you closer to an accurate and easier-to-manage result.

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