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Chapter 13: Problem 28

Simplify each expression. $$-\frac{1}{2}\left(6 d^{3}+4\right)+\left(8 d^{3}+3\right)$$

Short Answer

Expert verified
The simplified expression is \(5d^3 + 1\).

Step by step solution

01

Distribute the Negative Fraction

Start by distributing the \(-\frac{1}{2}\) to each term inside the parentheses: \[-\frac{1}{2}(6d^3 + 4) = -\frac{1}{2} \times 6d^3 - \frac{1}{2} \times 4 = -3d^3 - 2.\]
02

Write the Combined Expression

Rewrite the expression with the distributed terms: \[-3d^3 - 2 + (8d^3 + 3).\]
03

Combine Like Terms

Combine the like terms (the terms involving \(d^3\) and the constant terms): For \(d^3\) terms: \(-3d^3 + 8d^3 = 5d^3\).For constant terms:\(-2 + 3 = 1\).Thus, the expression becomes \[5d^3 + 1.\]
04

Final Simplified Expression

The final simplified form of the expression is \[5d^3 + 1.\]

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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 you often use without even realizing it. It's about taking a number or term outside of parentheses and multiplying it by each term inside. This can simplify expressions before combining like terms. In our exercise, we used the distributive property to handle the expression \(-\frac{1}{2}(6d^3 + 4)\). Here's how it works:
  • The number \(-\frac{1}{2}\) is multiplied by \(6d^3\), resulting in \(-3d^3\).
  • Next, \(-\frac{1}{2}\) is multiplied by 4, giving \(-2\).
Distributing the term simplifies expressions and prepares them for combining like terms. This step is crucial because it rearranges the expression into a form that's easier to simplify further.
Combining Like Terms
Combining like terms is a technique used to simplify expressions or equations. Like terms are terms that have the same variable raised to the same power. In our context, these are terms that can be added or subtracted from each other. Let's break it down:
  • In the expression \(-3d^3 - 2 + (8d^3 + 3)\), identify the like terms.
  • Terms \(-3d^3\) and \(8d^3\) both have the variable \(d^3\).
  • Add them together: \(-3d^3 + 8d^3 = 5d^3\).
  • For constants, \(-2 + 3 = 1\).
By combining these like terms, you further simplify the expression to \(5d^3 + 1\). It's a vital step that reduces the number of terms and presents a cleaner result.
Polynomials
Polynomials are expressions made up of terms which are usually composed of variables, coefficients, and exponents. They're foundational to algebra, helping to express complex problems simply. Here's why they're important:
  • A polynomial can have constants like 4 or variables like \(d^3\).
  • In our example, \(-\frac{1}{2}(6d^3 + 4) + (8d^3 + 3)\) is a polynomial with terms \(6d^3\), 4, \(8d^3\), and 3.
  • Understanding each term's role helps you identify how to distribute, combine, and simplify.
Polynomial terms can be rearranged and simplified through processes like distribution and combination, as seen in our finished expression \(5d^3 + 1\). Recognizing polynomials in equations helps in tackling a wide range of algebraic operations efficiently.

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