Problem 14 - Algebra A Combined Function

Question

Find each product and simplify if possible. See Examples 1 through 3. $$ \frac{a^{2}-4 a+4}{a^{2}-4} \cdot \frac{a+3}{a-2} $$

Step-by-Step Solution

Answer

The product simplifies to \( \frac{a+3}{a+2} \).

Step 1: Factor the Numerator of the First Fraction

The numerator of the first fraction is \( a^2 - 4a + 4 \). Recognize this as a perfect square trinomial: \( (a-2)^2 \). Thus, the numerator factors to \( (a-2)(a-2) \).

Step 2: Factor the Denominator of the First Fraction

The denominator of the first fraction is \( a^2 - 4 \). This expression is a difference of squares, which can be factored as \( (a-2)(a+2) \).

Step 3: Write Down the Second Fraction

The second fraction is \( \frac{a+3}{a-2} \), which is already in its simplest form and doesn't require any factoring.

Step 4: Multiply Numerators and Denominators

Multiply the numerators together: \((a-2)(a-2)(a+3)\).Multiply the denominators together: \((a-2)(a+2)(a-2)\).

Step 5: Cancel Common Factors

Identify and cancel out the common factor of \((a-2)\) in the numerator and \((a-2)\) in the denominator.The numerator becomes \((a-2)(a+3)\).The denominator becomes \((a+2)(a-2)\).

Step 6: Final Simplification

The fraction simplifies further to \( \frac{a+3}{a+2} \) after cancelling the remaining \(a-2\) in the numerator and denominator.

Key Concepts

Factoring Polynomials

Understanding how to factor polynomials is key to solving algebraic expressions. In the given exercise, we start by factoring the numerator and the denominator of the first fraction. This particular expression features a trinomial in the numerator: \( a^2 - 4a + 4 \). Trinomials like this are often perfect square trinomials, meaning they can be rewritten as the square of a binomial, \( (a-2)^2 \). Recognizing such patterns simplifies the process significantly.

The denominator, \( a^2 - 4 \), is a difference of squares. The difference of squares follows the pattern \( x^2 - y^2 = (x+y)(x-y) \). For \( a^2 - 4 \), we have \( a^2 - 2^2 \), which factors into \( (a-2)(a+2) \). Knowing these common factoring techniques enhances the ability to simplify complex polynomials effectively.

Rational Expressions

Rational expressions involve fractions where both the numerator and the denominator are polynomials. The key to working with them, especially in multiplication or division, is to factor where possible.

In our exercise, each part of the fraction must be checked and factored appropriately. The first fraction's numerator \((a-2)^2\) and denominator \((a-2)(a+2)\), after factoring, reveal parts that can be cancelled. This simplifies subsequent calculations.

The second fraction, \( \frac{a+3}{a-2} \), does not require factoring because it’s already in its simplest form. Multiplying rational expressions involves multiplying the numerators for the new numerator and the denominators for the new denominator. Keeping expressions in factored form helps identify common factors that simplify the expression.

Simplifying Fractions

Simplifying fractions is an essential step in working with rational expressions. By cancelling common factors, we reduce expressions to their simplest form. In our exercise, the common factor \( (a-2) \) is identified in both the numerator and the denominator.

Observing common factors:

• Write the expression in factored form.
• Identify any polynomial expressions that appear in both the numerator and denominator.
• Cancel out these common expressions to simplify.

After cancelling, if there are no remaining common factors, the expression is in its simplest form. In this example, after cancelling \( (a-2) \), the expression reduces to \( \frac{a+3}{a+2} \). Ensuring fractions are as simple as possible makes solving algebraic equations much more manageable.