Problem 27 - Elementary and Intermediate Algebra

Question

Multiply. See Example 2. $$ -4 t\left(t^{2}-7\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \(-4t^3 + 28t\).

Step 1: Understand the Distributive Property

The problem involves multiplying the term \(-4t\) by the expression in the parentheses, \(t^2 - 7\). This can be solved using the distributive property which states \(a(b + c) = ab + ac\).

Step 2: Apply the Distributive Property

Use the distributive property to multiply \(-4t\) with each term inside the parentheses, \(t^2\) and \(-7\). This results in: \(-4t \cdot t^2\) and \(-4t \cdot (-7)\).

Step 3: Simplify Each Product

Calculate the first product: \(-4t \cdot t^2\) gives \(-4t^3\), because the bases are the same and you add the exponents: \(t^1 \cdot t^2 = t^{1+2} = t^3\). For the second product: \(-4t \cdot (-7)\) equals \(28t\) because multiplying two negative numbers gives a positive result.

Step 4: Combine Simplified Products

Combine the two simplified expressions to get the answer. Therefore, \(-4t(t^2 - 7) = -4t^3 + 28t\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental component of mathematics, allowing us to represent numbers and operations efficiently. They consist of variables, constants, and arithmetic operations such as addition, subtraction, multiplication, and division. For example, in the algebraic expression \[-4t(t^2 - 7),\] we see a combination of all these elements. Variables (in this case, \(t\)) stand in for unknown values and can change based on the context or problem given. Constants are fixed values like \(-4\) and \(-7\) in this expression, which provide specific numeric values.
The use of variables allows algebraic expressions to model real-world situations where certain quantities are unknown or changeable. Additionally, writing expressions simplifies the representation of complex mathematical scenarios. Understanding the structure of algebraic expressions is essential as they form the basis for more complicated equations and operations in mathematics.

Polynomial Multiplication

Polynomial multiplication involves multiplying expressions that are made up of multiple terms. A polynomial is essentially an expression that can have constants, variables, and exponents. In our problem, we're multiplying a monomial, \(-4t\), by a binomial, \(t^2 - 7\). This requires applying the distributive property, which states that every term in the first expression needs to be multiplied by every term in the second.To carry out this multiplication, you

• First, take \(-4t\) and multiply it by the first term in the parenthesis \(t^2\), resulting in \(-4t^3\).
• Next, multiply \(-4t\) by the second term, \(-7\), which results in \(28t\), since a negative times a negative yields a positive.

After performing these multiplications, you combine the results to keep the expression simplified yet accurate.
Using these rules ensures the multiplication process is systematic and the polynomial correctly expanded.

Simplifying Expressions

Simplifying expressions is the process of making an expression into its most straightforward form while preserving its value. It involves combining like terms, reducing operations, and ensuring the expression is as compact as possible. For the expression resulting from our multiplication, \(-4t^3 + 28t\), we check if further simplification is possible:

• First, look for like terms to combine. Like terms have exactly the same variable part. Here, \(-4t^3\) and \(28t\) are not like terms since their variable parts \(t^3\) and \(t\) differ.
• Since there are no like terms, \(-4t^3 + 28t\) is already in its simplest form.

Being able to simplify effectively helps clarify algebraic expressions and prepare them for further calculations or for solving equations. It’s a critical skill that enables easier interpretation and manipulation of expressions in problem-solving scenarios.