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Chapter 1: Problem 90

Simplify. $$ 4\left(d^{2}-3\right)-\left(d^{2}-1\right) $$

Short Answer

Expert verified
The simplified expression is \(3d^2 - 11\).

Step by step solution

01

Expand the expressions

Begin by distributing any coefficients outside of the parentheses to the terms inside the parentheses. For the first part, distribute the 4: \[ 4(d^2 - 3) = 4d^2 - 12 \] For the second part, distribute the -1: \[ -(d^2 - 1) = -d^2 + 1 \]
02

Combine like terms

Now that the expression has been expanded, combine the like terms:\[ 4d^2 - 12 - d^2 + 1 \]Combine the \(d^2\) terms:\[ 4d^2 - d^2 = 3d^2 \]And the constant terms:\[ -12 + 1 = -11 \]
03

Write the simplified expression

Collect the simplified terms to write the final simplified version of the original expression:\[ 3d^2 - 11 \]

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Key Concepts

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

Combining Like Terms
When you come across the phrase "combining like terms," it refers to the process of simplifying an expression by gathering terms that have the same variable raised to the same power. For example, in the expression \(4d^2 - d^2\), both terms are "like terms" because they have the same variable, \(d\), raised to the same exponent, 2.

Here's how you can easily combine like terms:
  • Look for terms that have the same variable and exponent.
  • Add or subtract their coefficients, which are the numbers in front of the variables.
In our exercise, when combining \(4d^2\) and \(-d^2\), you simply subtract 1 from 4, getting \(3d^2\).

This process not only simplifies the expression, making it easier to work with, but is also a crucial step in algebra to solve equations effectively and keep your work neat.
Distributive Property
The distributive property is a fundamental algebraic principle used to simplify expressions involving parentheses. When you see an expression like \(4(d^2 - 3)\), the distributive property allows you to multiply the 4 with each term inside the parentheses. This is done as follows:
  • Multiply the coefficient (in this case, 4) by each term within the parentheses.
  • Ensure you distribute to all terms inside the parentheses, respecting the order of operations.
Using our example, you multiply 4 by \(d^2\) to get \(4d^2\) and by \(-3\) to get \(-12\).

The same principle applies when distributing negative coefficients, such as in \(-(d^2 - 1)\). Distribute the \(-1\) to obtain \(-d^2\) and \(+1\).

Mastering the distributive property can effectively dismantle complex expressions, preparing them for further simplification steps like combining like terms.
Simplifying Expressions
Simplifying expressions is the process of rewriting an expression so it's as simple as possible, without changing its value. It usually involves applying multiple algebraic principles, like the distributive property and combining like terms, to break down complex expressions.
  • Start with applying the distributive property to remove parentheses.
  • Combine all like terms.
  • Ensure that your final expression has no like terms left uncombined.
In the given exercise, after distributing and combining like terms, we reach a final expression of \(3d^2 - 11\).

Simplifying makes equations and algebraic expressions easier to solve and understand. It provides clarity when working through algebra problems and helps to ensure accuracy in solutions. Always remember, the goal is to make the expression as straightforward and manageable as possible.

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Most popular questions from this chapter

To determine the average afternoon wait time in security lines at an airport, officials monitored four passengers, each at a different gate. The time that each passenger entered a security line and the time the same passenger cleared the checkpoint was recorded, as shown below. Find the average (mean) wait time for these passengers. $$ \begin{array}{|l|c|c|} \hline & {\text { Time }} & {\text { Time }} \\ \hline & {\text { entered }} & {\text { cleared }} \\ \hline \text { Passenger at Gate A } & {3: 05 \text { pm }} & {3: 21 \text { pm }} \\ \hline \text { Passenger at Gate B } & {3: 03 \text { pm }} & {3: 13 \text { pm }} \\ \hline \text { Passenger at Gate C } & {3: 01 \text { pm }} & {3: 09 \text { pm }} \\ \hline \text { Passenger at Gate D } & {3: 02 \text { pm }} & {3: 16 \text { pm }} \\ \hline \end{array} $$

Simplify each expression, if possible. $$ -1.1 m-(-2.4 m) $$

Simplify each expression, if possible. $$ 4 a+4 b+4 c $$

Simplify each expression, if possible. a. \(-3(-4 a)(-2)\) b. \(-3(-4 a)-2\)

Simplify each expression, if possible. $$ 10-5(-5 g-1) $$
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