Problem 62 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ (4 t+0.6)(4 t-0.6) $$

Step-by-Step Solution

Answer

The expression simplifies to \(16t^2 - 0.36\).

Step 1: Identify the Expression

We begin with the expression \((4t + 0.6)(4t - 0.6)\). This is in the form of \((a + b)(a - b)\), where \(a = 4t\) and \(b = 0.6\).

Step 2: Recall the Difference of Squares Formula

The expression is recognized as a difference of squares, where \((a + b)(a - b) = a^2 - b^2\).

Step 3: Apply the Formula

Using the formula for the difference of squares, \(a = 4t\) and \(b = 0.6\), so we have:\[(4t)^2 - (0.6)^2\]

Step 4: Calculate the Squares

Now, calculate each square:1. \((4t)^2 = 16t^2\)2. \((0.6)^2 = 0.36\)

Step 5: Subtract the Values

Subtract the square of \(b\) from the square of \(a\):\[16t^2 - 0.36\]

Step 6: Combine and Simplify

Combine the terms to write the final simplified expression:\[16t^2 - 0.36\]

Key Concepts

Algebraic Identities

Algebraic identities are mathematical expressions that are true for all values of the variables involved. These identities serve as the backbone for many algebraic operations, simplifying complex expressions. One of the most significant algebraic identities is the difference of squares, represented by

• \((a + b)(a - b) = a^2 - b^2\)

This identity is particularly helpful because it allows us to transform a multiplication problem into a simpler subtraction problem. By recognizing patterns such as the difference of squares among variables, calculations become much more straightforward. Students should familiarize themselves with these identities to enhance their ability to efficiently simplify polynomial expressions.

Binomial Expressions

Binomial expressions involve two terms, typically joined by a plus or minus sign. In algebra, binomials play a crucial role in polynomial operations, as they often appear in factorizations and expansions. A typical example of a binomial expression is

• \((4t + 0.6)\)

Knowing how to work with binomials is essential, as they frequently appear in algebraic equations and expressions.

When binomials are multiplied together, as seen in the exercise

• \((4t + 0.6)(4t - 0.6)\)

they can sometimes form a pattern that corresponds to an algebraic identity. Recognizing these patterns can dramatically simplify the task of expanding and solving the expressions. Mastery of binomial expressions enables students to tackle more complex algebra problems with confidence.

Polynomial Operations

Polynomial operations encompass a range of mathematical procedures involving the addition, subtraction, multiplication, and division of polynomials. These operations form the foundation of algebra and are crucial for simplifying and solving equations.

Understanding how to manipulate polynomials is key. When working with a product of binomials, such as in our initial problem, applying known algebraic identities like the difference of squares allows us to conduct polynomial operations efficiently. By rewriting the expression

• \((4t)^2 - (0.6)^2\)

we reduce the problem to simple subtraction.

Students should practice these operations to develop a comprehensive understanding of polynomial behavior and the algebraic identities that guide them. Mastery of these concepts is essential for progressing to more advanced mathematical topics.