Problem 27 - Precalculus Mathematics for Calculus

Question

Perform the multiplication or division and simplify. $$\frac{t-3}{t^{2}+9} \cdot \frac{t+3}{t^{2}-9}$$

Step-by-Step Solution

Answer

Simplified expression is \( \frac{1}{t^2+9} \).

Step 1: Write the Expression

Write the expression that needs to be simplified: \( \frac{t-3}{t^{2}+9} \cdot \frac{t+3}{t^{2}-9} \).

Step 2: Factoring Denominator

Notice that the denominator \( t^2 - 9 \) can be factored using the difference of squares. Thus, \( t^2 - 9 = (t-3)(t+3) \).

Step 3: Rewrite with Factored Denominator

Rewrite the expression using the factored form: \( \frac{t-3}{t^{2}+9} \cdot \frac{t+3}{(t-3)(t+3)} \).

Step 4: Cancel Common Factors

Identify and cancel the common terms \( t-3 \) and \( t+3 \) from the numerators and denominators: \( \frac{1}{t^{2}+9} \cdot \frac{1}{1} \).

Step 5: Simplified Expression

Multiply the remaining fractions: \( \frac{1}{t^{2}+9} \). This is the simplified form of the expression.

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in algebra that involves breaking down a polynomial into simpler, more manageable components or factors. This process is like unboxing a complex expression, revealing particular components that are easier to handle. For example, consider the polynomial expression given in the exercise. We have a term in one of the denominators: \( t^2 - 9 \).Here, the goal is to factor this quadratic expression. Polynomials like these can often be broken down using various methods, one of which is the difference of squares. By factoring, we make the expression easier to simplify. Understanding how to factor polynomials is crucial because:

• It allows you to simplify expressions that may otherwise seem complicated.
• It is a stepping stone to solving polynomial equations.
• It helps in finding the roots or zeroes of the polynomial.

So, whenever you see a polynomial, always consider if it can be simplified by factoring it into smaller, simpler polynomials.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Simplifying rational expressions is a fundamental part of algebra, where the objective is to reduce the expression to its lowest terms.In the initial exercise, the given expression was a rational expression. To simplify such expressions:

• Factor both the numerator and the denominator wherever possible.
• Cancel out the common factors from both the numerator and the denominator.

For instance, simplifying \( \frac{t-3}{t^2+9} \cdot \frac{t+3}{(t-3)(t+3)} \) involves identifying common factors after factoring polynomials, such as \((t-3)\) and \((t+3)\).Once these common factors are canceled, the expression significantly simplifies, as seen when our initial complex expression results in \( \frac{1}{t^2+9} \). Remember, a simplified rational expression will not have any removable factors reappear in this reduced form.

Difference of Squares

The difference of squares is a specific method used to factor certain polynomials. It applies specifically to expressions such as \( a^2 - b^2 \), and it factors neatly into \( (a-b)(a+b) \).In the exercise provided, recognizing that the term \( t^2 - 9 \) is a difference of squares allows us to factor it into \((t-3)(t+3)\). This simplifies the rational expression significantly and is a key concept in reducing complex algebraic fractions. The difference of squares is crucial when you encounter:

• A subtraction between two perfect squares.
• Quadratic expressions that don't fit into linear factoring methods.

By understanding and applying the difference of squares, you unlock a powerful tool in algebra that simplifies polynomials by recognizing inherent patterns, leading to more efficient problem-solving.