Problem 21 - Algebra 2

Question

Simplify each expression. \(\frac{9-t^{2}}{t^{2}+t-12}\)

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{(3-t)(3+t)}{(t+4)(t-3)}\).

Step 1: Factor the Numerator

First, we need to factor the numerator expression \(9 - t^2\). Notice that this is a difference of squares, which can be factored using the formula \(a^2 - b^2 = (a-b)(a+b)\). Here, \(9 - t^2\) can be rewritten as \((3)^2 - (t)^2\), which factors to \((3 - t)(3 + t)\).

Step 2: Factor the Denominator

Next, we factor the denominator \(t^2 + t - 12\). We look for two numbers that multiply to \(-12\) and add to \(1\). These numbers are \(4\) and \(-3\). So, we can write \(t^2 + t - 12\) as \((t + 4)(t - 3)\).

Step 3: Simplify the Fraction

With both the numerator and denominator factored, the expression becomes \(\frac{(3 - t)(3 + t)}{(t + 4)(t - 3)}\). Observe that there's no common factor in the numerator and the denominator, so no further simplification can be made.

Key Concepts

Factoring

Factoring is the process of breaking down complex algebraic expressions into simpler components or products of factors. These simpler parts are often easier to handle and work with in various algebraic operations. Understanding factoring is essential in simplifying algebraic expressions and solving equations.

When we factor something, we look for expressions that multiply to give back the original term. Basic factoring involves identifying a common factor among terms or using specific formulas to rewrite expressions. For instance, expressions like difference of squares and trinomials require specific techniques.

• In our example, we factored both the numerator and denominator. The numerator is factored as a difference of squares, while the denominator is factored as a trinomial.
• Recognizing patterns in expressions allows for quicker and more efficient factoring.

Mastering the various methods of factoring, such as grouping, using identities, or trial and error, is a significant step in advancing your algebra skills.

Difference of Squares

The difference of squares is a special kind of polynomial that takes advantage of the formula: \(a^2 - b^2 = (a - b)(a + b)\). This formula helps simplify expressions where two perfect squares are subtracted from one another.

In our original exercise, the expression in the numerator, \(9 - t^2\), is a classic example of a difference of squares. In this case:

• \(a = 3\) and \(b = t\)
• We rewrite \(9 - t^2\) as \((3)^2 - (t)^2\)
• Factoring this using the formula gives us \((3 - t)(3 + t)\)

Identifying a difference of squares quickly lets you simplify an expression by breaking it down into linear factors, which can be more easily combined or cancelled when simplifying fractions or solving equations.

Simplifying Fractions

Simplifying fractions in algebra involves reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. This process often entails factoring both the numerator and the denominator to identify and cancel out such common factors.

For the expression given in the original exercise, after factoring the numerator and denominator, we obtained:

• Numerator: \((3 - t)(3 + t)\)
• Denominator: \((t + 4)(t - 3)\)

When simplifying algebraic fractions:

• Check for common factors between the numerator and the denominator.
• In this example, there were no factors to cancel, so the fraction remained the same.
• Remember, a simplified fraction is easier to work with in further algebraic calculations.

Mastering the simplification process is crucial for efficient problem solving and ensures that you understand the relationships between the terms in the fractions.