Problem 4 - College Algebra

Question

Simplify the rational expressions. $$ \frac{x^{2}-16}{x^{2}-5 x+4} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x+4}{x-1}\).

Step 1: Factor the Numerator

The numerator is a difference of squares: \(x^2-16\). Using the formula \(a^2-b^2 = (a-b)(a+b)\), we can factor this as \((x-4)(x+4)\). So, the numerator becomes \((x-4)(x+4)\).

Step 2: Factor the Denominator

The denominator is a quadratic expression \(x^2-5x+4\). We need to find two numbers that multiply to 4 and add to -5. These numbers are -4 and -1. So, the denominator factors as \((x-4)(x-1)\).

Step 3: Cancel Common Factors

Now that we have the factors, the expression is \(\frac{(x-4)(x+4)}{(x-4)(x-1)}\). We can cancel the common factor \((x-4)\) in both the numerator and the denominator.

Step 4: Write the Simplified Expression

After canceling the common factor, the simplified expression is \(\frac{x+4}{x-1}\).

Key Concepts

Factoring Quadratics

Factoring quadratics is a key skill in simplifying rational expressions. A quadratic expression is typically in the form of \(ax^2 + bx + c\). In many cases, you need to factor the expression to simplify a rational expression or solve an equation, as seen in the exercise's denominator \(x^2-5x+4\).

To factor a simple quadratic, look for two numbers that:

• Multiply to give you the constant term (\(c\))
• Add to give you the middle coefficient (\(b\))

In our case, these numbers are -4 and -1:

• \(-4 \times -1 = 4\)
• \(-4 + -1 = -5\)

Once you identify these numbers, you can express the quadratic as the product of two binomials. So, \(x^2-5x+4\) factors into \((x-4)(x-1)\). Remember, practice makes perfect and understanding this concept allows you to tackle more complex expressions.

Difference of Squares

The difference of squares is a specific type of factoring used when a quadratic takes on the form \(a^2 - b^2\). This pattern is special because it can always be expressed as two binomials multiplied together: \((a-b)(a+b)\).

In the numerator of our exercise, \(x^2-16\), we recognize this form where \(a=x\) and \(b=4\). Here, 16 is \(4^2\), so we have:

• \(x^2 - 4^2\)
• \((x-4)(x+4)\)

This method quickly breaks down complex expressions into simpler pieces you can work with. Using the difference of squares offers a straightforward way to simplify the initial expression before moving on to further steps. Identifying and applying this can often make seemingly daunting problems much easier to solve.

Canceling Common Factors

Canceling common factors is the final crucial step in simplifying rational expressions. After factoring both the numerator and denominator, you’ll often find common terms that can be deleted, simplifying the expression significantly.

In our exercise, after expanding the expression into \((x-4)(x+4)\) over \((x-4)(x-1)\), we observe that \((x-4)\) appears in both the numerator and denominator. This shared factor can be canceled, leaving you with the simplified expression \(\frac{x+4}{x-1}\).

Remember:

• Only cancel terms that are factors of both the numerator and the denominator, not terms that are summed or subtracted.
• Check your work to ensure that canceling any factor doesn’t change the domain of the expression unnecessarily (e.g., \(x=4\) cannot be in the domain).

Canceling common factors makes the expression more manageable and often reveals a simplified form that’s easier to interpret or analyze. This step shows the power of recognizing and utilizing patterns in algebra.