Problem 41 - Intermediate Algebra

Question

Simplify each rational expression. $$ \frac{4 a^{2}-25}{20 a-50} $$

Step-by-Step Solution

Answer

Simplified expression: \( \frac{2a + 5}{10} \).

Step 1: Identify the structure of the expression

The expression is a fraction with a polynomial in the numerator and another in the denominator. It reads as \( \frac{4a^2 - 25}{20a - 50} \). The goal is to simplify this expression by factoring whenever possible.

Step 2: Factor the numerator

The numerator \( 4a^2 - 25 \) is a difference of squares. Recall that \( a^2 - b^2 = (a+b)(a-b) \). Here \( 4a^2 - 25 \) can be rewritten as \( (2a)^2 - (5)^2 \), resulting in \((2a + 5)(2a - 5) \).

Step 3: Factor the denominator

The denominator \( 20a - 50 \) can be simplified by factoring out the greatest common factor, which is 10: \( 20a - 50 = 10(2a - 5) \).

Step 4: Cancel common factors

Now, we have the expression \( \frac{(2a + 5)(2a - 5)}{10(2a - 5)} \). Observe that \(2a - 5\) appears in both the numerator and the denominator. Cancel the common factor \( (2a - 5) \).

Step 5: Write the simplified expression

After canceling \( (2a - 5) \), the expression simplifies to \( \frac{2a + 5}{10} \). This is the simplest form of the original rational expression.

Key Concepts

Factoring Polynomials

Polynomials can often look intimidating, but factoring them makes things much clearer. Factoring essentially means breaking down a polynomial into simpler, multiplied components. Think of it like pulling apart a whole into bite-sized pieces.
For example, consider the polynomial in the numerator of our problem:

• The goal here is to represent the polynomial as a product of smaller polynomials.
• In the case of the numerator, the polynomial is written as \(4a^2 - 25\).
• Upon closer inspection, this can be expressed as a "difference of squares" and factored into \((2a + 5)(2a - 5)\).

Making sure to accurately factor each polynomial is key. It’s a tool that simplifies complex expressions, making them easier to work with. And, once broken down, it becomes a stepping stone to further simplify rational expressions.

Difference of Squares

The difference of squares is a special type of polynomial. Recognizing this pattern allows you to quickly factor and simplify expressions.
Here's how it works:

• The formula for the difference of squares is: \(a^2 - b^2 = (a+b)(a-b)\).
• Our example includes the expression \(4a^2 - 25\). Here, notice that \(4a^2\) is \((2a)^2\) and \(25\) is \(5^2\).
• Applying the formula, we rewrite it as \((2a + 5)(2a - 5)\).

Recognizing such patterns saves time and effort. It gives one a shortcut to simplify expressions without laborious trial and error.

Greatest Common Factor

Finding the greatest common factor (GCF) is like identifying a common thread running through terms in a polynomial. This shared factor is the biggest number or expression that divides each term evenly.
Consider the denominator in the original problem:

• The expression \(20a - 50\) can be simplified by first identifying that both terms share the factor 10.
• Dividing each term by 10, we factor the expression to \(10(2a - 5)\).

This step is crucial because it sets the stage for further simplification. Once you reduce a polynomial by its GCF, it becomes more manageable and often reveals further factoring possibilities.