Problem 37 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{x+5}{x^{2}-4 x-45} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{1}{x-9}\), for \(x \neq -5\).

Step 1: Factor the Denominator

The expression in the denominator is a quadratic trinomial in the form of \(x^2 - 4x - 45\). To factor this expression, find two numbers that multiply to -45 and add to -4. These numbers are -9 and 5. Thus, we can factor the denominator as \((x - 9)(x + 5)\).

Step 2: Simplify the Fraction

The original fraction is \(\frac{x+5}{(x-9)(x+5)}\). Notice that the numerator, \(x+5\), is a common factor in both the numerator and denominator. Cancel out the common factor \(x+5\) from the numerator and the denominator.

Step 3: Write the Simplified Expression

After canceling \(x+5\), the expression simplifies to \(\frac{1}{x-9}\). Make sure to note that the simplification is valid for \(x eq -5\) because \(x+5\) would make the original denominator zero.

Key Concepts

Factoring Quadratics

Factoring quadratics is like guessing a mystery number based on two clues. You know a quadratic equation has the form \(ax^2 + bx + c\). In the given problem, the quadratic is \(x^2 - 4x - 45\), where \(a = 1\), \(b = -4\), and \(c = -45\). The goal is to break this into two simple factors that look like \((x + m)(x + n)\).

To find \(m\) and \(n\), search for two numbers that multiply to \(-45\) (the constant term) and add up to \(-4\) (the coefficient of \(x\)). These numbers are \(-9\) and \(5\).

• - Multiplies: \((-9)\times(5) = -45\)
• - Adds: \(-9 + 5 = -4\)

Therefore, \(x^2 - 4x - 45\) is factored into \((x - 9)(x + 5)\). Factoring makes complex expressions more manageable, simplifying future steps.

Canceling Common Factors

Canceling common factors in algebraic expressions is similar to reducing fractions in arithmetic. You simplify by removing common parts in the numerator and the denominator.

In the given expression \(\frac{x+5}{(x-9)(x+5)}\), both the top and bottom share \(x+5\). You can cancel out this repeated factor. Why? Because dividing any term by itself, as long as it is not zero, results in one.

• - Original: \(\frac{x+5}{(x-9)(x+5)}\)
• - After Canceling \(x+5\): \(\frac{1}{x-9}\)

But be cautious! The canceled term \(x+5\) must not be zero. Hence, the fraction hugs the condition \(x eq -5\) to stay valid, because \(x = -5\) makes both the denominator and numerator zero in the original expression.

Rational Expressions

Rational expressions are like fractions, but with polynomials. They involve a numerator and a denominator which are each polynomials. The challenge is to manipulate them as with fractions, keeping in mind the rules of algebra.

For rational expressions, the operation of simplification involves factoring and canceling, as seen in our example \(\frac{x+5}{x^2-4x-45}\). Begin by expressing each polynomial in its most factorized form. Next, cancel any common factors.

• - Original: \(\frac{x+5}{x-9)(x+5)}\)
• - Simplified: \(\frac{1}{x-9}\)

One must always be attentive to the conditions under which these simplifications hold. For instance, in the given expression, the simplification lacks validity if \(x = -5\) due to undefined values in the original fraction.

Rational expressions require extra attention for the defined domains, ensuring no division by zero occurs.