Problem 42 - Beginning and Intermediate Algebra

Question

Factor completely. $$t^{2}-\frac{1}{100}$$

Step-by-Step Solution

Answer

The completely factored form of the given expression is \((t + \frac{1}{10})(t - \frac{1}{10})\).

Step 1: Identify the terms a and b

In this expression, we can see that \(a^2 = t^2\) and \(b^2 = \frac{1}{100}\). So, we have \(a = t\) and \(b = \frac{1}{10}\) because \(\sqrt{t^2}=t\) and \(\sqrt{\frac{1}{100}}=\frac{1}{10}\).

Step 2: Apply the difference of two squares rule

Using the difference of two squares rule \((a+b)(a-b)\), substitute the values of a and b that we found in step 1. This gives us \((t + \frac{1}{10})(t - \frac{1}{10})\).

Step 3: Write the final factored form

Therefore, the completely factored form of the given expression is \((t + \frac{1}{10})(t - \frac{1}{10})\).

Key Concepts

Difference of Squares

The concept of "difference of squares" is fundamental in algebra, especially when factoring polynomials. It refers to an expression in the form \(a^2 - b^2\). This specific structure is unique because it can be rewritten using the formula:\[(a + b)(a - b)\]This formula emerges from expanding the product of \((a + b)\) and \((a - b)\), which shows that the middle terms cancel each other out, leaving us with the difference of squares. Knowing this helps us factor expressions quickly without having to resort to lengthy calculations.
For example, the expression \(t^2 - \frac{1}{100}\) is a difference of squares because both terms are perfect squares: \(t^2\) and \(\left( \frac{1}{10} \right)^2\). By identifying these terms as \(a = t\) and \(b = \frac{1}{10}\), we can directly substitute them into the difference of squares formula to achieve the factored form:\[(t + \frac{1}{10})(t - \frac{1}{10})\]Realizing when an expression is a difference of squares helps simplify what might initially seem daunting.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. They represent mathematical situations and can be manipulated to simplify or solve problems.
Key components of an algebraic expression include:

• Terms: Parts of the expression separated by addition or subtraction.
• Variables: Symbols that represent unknown values, commonly \(x, y, t\), etc.
• Coefficients: The numbers multiplying the variables.
• Constants: Numbers without variables.

In the expression \(t^2 - \frac{1}{100}\), "\(t\)" is the variable, and it's raised to the power of 2, making the first term a "monomial." The second term, "\(-\frac{1}{100}\)," is a constant.
The entire expression contains two terms, making it a "binomial."
Understanding the structure of algebraic expressions allows you to apply appropriate mathematical techniques, like factoring, effectively.

Quadratic Equations

Quadratic equations are a type of polynomial equation represented in the general form:\[ax^2 + bx + c = 0\]where \(a\), \(b\), and \(c\) are constants, with \(a eq 0\). These equations surface in many mathematical contexts and require specific methods to solve.
While the expression \(t^2 - \frac{1}{100}\) is not set equal to anything (an equation form), it is related to the concept of solving quadratic equations because it involves a squared term.
If as part of a problem, it were set equal to zero, it would take the form of a quadratic equation \[t^2 - \frac{1}{100} = 0\]This could be solved by factoring, using the difference of squares method, or using other techniques like completing the square or the quadratic formula.
Understanding how to switch expressions into equations and solve them is a key aspect of working with quadratics.