Problem 25 - Beginning and Intermediate Algebra

Question

Divide. $$\frac{k^{2}+4 k-45}{k+9}$$

Step-by-Step Solution

Answer

The simplified rational expression is \(k - 5\).

Step 1: Factor the numerator

First, we'll look to see if the numerator, \(k^2 + 4k - 45\), can be factored. If it can, we will factor it and then move to the next steps. To factor the numerator, we are looking for two factors that multiply to -45 and add to 4. Those factors are (k + 9) and (k -5). So, the numerator can be written as: \((k + 9)(k - 5)\)

Step 2: Insert factored numerator into the expression

Now we will insert the factored numerator into the given expression. So we have: \(\frac{(k + 9)(k - 5)}{k + 9}\)

Step 3: Cancel the common factor

Now, we'll cancel out the common factor in both the numerator and denominator. In this case, the common factor is (k + 9): \(\frac{(k + 9)(k - 5)}{k + 9} = \frac{\cancel{(k + 9)}(k - 5)}{\cancel{k + 9}}\)

Step 4: Simplify

Simplifying the canceled rational expression, we are left with \(k - 5\) So our final answer is: \(k - 5\)

Key Concepts

Factoring Quadratic Expressions

Factoring quadratic expressions is an essential skill when working with polynomials and rational expressions. A quadratic expression is typically of the form \( ax^2 + bx + c \). The goal is to express it as a product of two binomials. To factor such expressions, you need to find two numbers that multiply to the constant term, \( c \), and add up to the linear coefficient, \( b \).
In our example, the quadratic expression is \( k^2 + 4k - 45 \). We look for two numbers that multiply to \(-45\) (the constant term) and sum to \(4\) (the coefficient of \( k \)). After some consideration, we find that the numbers \( 9 \) and \( -5 \) satisfy this condition.
Thus, the expression \( k^2 + 4k - 45 \) can be factored into \((k + 9)(k - 5)\). Factoring is a crucial first step in simplifying expressions as it often reveals the common factors allowing further simplification.

Canceling Common Factors

Once a polynomial is factored, you may often notice that there are common factors present in both the numerator and the denominator. Identifying and canceling these common factors is crucial for simplifying rational expressions. In the case of our example, after factoring the numerator \( k^2 + 4k - 45 \) into \((k + 9)(k - 5)\), we see the numerator \((k + 9)(k - 5)\) and the denominator \( k + 9 \) both contain the factor \( k + 9 \).
Canceling involves eliminating these common terms, as they divide evenly, effectively removing them from the expression.

• Write the fraction with factored terms.
• Identify identical bins in both the numerator and the denominator.
• Cross out or 'cancel' those common factors.

This leaves you with a simplified expression, which in this case is \( k - 5 \). It's crucial to ensure that the canceled factors do not equal zero, as this would make the original rational expression undefined.

Simplifying Rational Expressions

Simplifying rational expressions entails reducing the complexity of the expression while retaining the same value for its defined domain. After you have factored and canceled any common factors, you can be left with a much more straightforward expression.
The simplified form of the expression from our example is \( k - 5 \). This result comes from first factoring and then canceling the \( k + 9 \) terms that appear in both the numerator and the denominator.
Simplifying rational expressions helps in understanding the behavior of the expression across its domain. It gives easier and clearer results, especially when solving or analyzing equations and inequalities. Always remember:

• Check the original expression and note any restrictions, such as values that would make the denominator zero.
• Factor wherever possible.
• Cancel common factors carefully ensuring you account for any restrictions.

This process of simplifying rational expressions is fundamental in algebra and higher-level mathematics.