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Chapter 1: Problem 107

Simplify. $$ -7 t^{3}-t^{2}-3\left(5 t^{3}-3 t\right) $$

Short Answer

Expert verified
-22t^3 - t^2 + 9t

Step by step solution

01

Distribute the Multiplication

Distribute the -3 across the terms inside the parenthesis. Multiply -3 by each term inside the parenthesis: \(-3(5t^3 - 3t) = -3 \times 5t^3 - 3 \times -3t\). This simplifies to \(-15t^3 + 9t\).
02

Combine Like Terms

Combine the like terms from the original equation and the distributed terms. The original expression is \(-7t^3 - t^2 - 15t^3 + 9t\). Combine the \(-7t^3\) term with the \(-15t^3\) term to get \(-22t^3\). The \(-t^2\) term remains the same, and the \(+9t\) term remains as well.

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Key Concepts

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

Distributive Property
To simplify the given expression, we first need to use the distributive property. This property allows us to distribute the multiplication of a single term across the terms inside a parenthesis.
In algebra, the distributive property is represented as: \[ a(b + c) = ab + ac \]
Applying this to our exercise, we distribute \( -3 \) over the terms inside the parentheses in \( -3(5t^3 - 3t) \).
So:
\( -3 \times 5t^3 - (-3) \times 3t = -15t^3 + 9t \).
This step helps in breaking down the expression for easier simplification.
Combining Like Terms
After distributing the terms, we proceed to combine like terms in the expression. 'Like terms' are terms that have the same variable raised to the same power.
Let's consider the following expression: \( -7t^3 - t^2 - 15t^3 + 9t \).
Identify the like terms:
    \r
  • -7t^3 and -15t^3 (both are t^3 terms)\r
    • \r
  • -t^2 (there is no other t^2 term)\r
    • \r
  • 9t (there is no other t term)\r
    • \r
Sum up the like terms:
    \r
  • Combine the t^3 terms: \( -7t^3 - 15t^3 = -22t^3 \)\r
    • \r
  • The t^2 term remains the same: \( -t^2 \)\r
    • \r
  • The t term remains the same: \( +9t \)\r
    • \r
This simplifies our expression to: \( -22t^3 - t^2 + 9t \).
Combining like terms is crucial to reach a simple, understandable form of algebraic expressions.
Algebraic Expressions
An algebraic expression consists of variables, coefficients, and constants, connected by arithmetic operations like addition, subtraction, multiplication, and division.
In our context, the expression \( -7t^3 - t^2 - 3(5t^3 - 3t) \) includes:
    \r
  • Variables: \( t \) \r
    • \r
  • Coefficients: Numbers multiplying the variables (e.g., -7, -3, 5, -3)\r
    • \r
  • The constants are implied within the coefficients set to the variables\r
    • \r
Algebraic expressions form the basis of algebra, letting us model real-world problems.
When simplifying them, following steps like the distributive property and combining like terms ensure clarity and proper solution.

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Most popular questions from this chapter

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