Problem 28 - Elementary and Intermediate Algebra

Question

Find each product. See Example 2. $$ (5 t+4)(5 t-4) $$

Step-by-Step Solution

Answer

The product is \(25t^2 - 16\).

Step 1: Recognize the Formula

Identify the algebraic structure in the expression \((5t+4)(5t-4)\). This is a difference of squares, where \(a = 5t\) and \(b = 4\). Recall that the difference of squares formula is \((a + b)(a - b) = a^2 - b^2\).

Step 2: Apply the Difference of Squares Formula

Apply the formula to the expression: \((a + b)(a - b) = a^2 - b^2\). In this case, substitute \(a = 5t\) and \(b = 4\) into the formula: \((5t)^2 - 4^2\).

Step 3: Calculate Each Square

First calculate \((5t)^2\) which is equivalent to \(5^2 \cdot t^2 = 25t^2\). Then calculate \(4^2\) which equals \(16\).

Step 4: Compute the Final Product

Subtract the second square from the first: \(25t^2 - 16\). This is the simplified result of the expression \((5t+4)(5t-4)\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators that together represent a mathematical relationship or quantity. In this context, expressions can be simple, like a single term, or more complex, as in polynomials. Understanding algebraic expressions is crucial for solving various types of mathematical problems. Here’s what to remember about algebraic expressions:

• Terms: These are the building blocks of expressions. Each term consists of a coefficient (a numerical value) and one or more variables raised to a power.


• Variables: Symbols that represent unknown values in expressions and equations, often denoted by letters like \(t\), \(x\), or \(y\).


• Operators: Symbols such as \(+\), \(-\), \(\times\), and \(\div\) used to combine terms in expressions.

Understanding these elements helps to decipher complex expressions, allowing students to simplify, solve, and manipulate expressions to find solutions.

Multiplying Polynomials

Multiplying polynomials involves combining each term from one polynomial with every term in another polynomial. This process results in a new polynomial. The key is to use distributive property, often required when dealing with larger expressions:

• Distributive Property: Apply each term of one polynomial to every term of the other. It ensures all possible products of terms are considered, resulting in the final expanded polynomial.


• Common Mistakes: Forgetting to multiply each term by the others or neglecting correct exponent rules can lead to incorrect products.

In our case, - Recognizing the structure as a difference of squares simplifies the process immensely. - The formula \((a + b)(a - b) = a^2 - b^2\) quickens the multiplication by transforming it directly to difference without individual term expansion. Understanding these concepts allows efficient multiplication of expressions like polynomials with ease.

Simplifying Expressions

Simplifying expressions is the practice of rewriting them in a more manageable or reduced form, making them easier to understand and solve. Simplification often involves combining like terms or reducing fractions and expressions:

• Combine Like Terms: Terms with the same variable and power should be combined to simplify the expression.


• Use of Formulas: Utilization of algebraic identities such as the difference of squares can facilitate simplification. - As seen with \((5t+4)(5t-4)\), applying \(a^2 - b^2\) reduces the expression swiftly to a cleaner \(25t^2 - 16\).

Removing complexity by reducing expressions helps solve equations faster and ensures understanding of the underlying relationships within expressions.