Problem 43 - Elementary Algebra

Question

For exercises 1-66, simplify. $$ \frac{k^{2}-3 k-40}{k^{2}-4 k-45} $$

Step-by-Step Solution

Answer

\( \frac{k - 8}{k - 9} \)

Step 1: Factor the Numerator

The numerator of the fraction is \text{k^{2}-3k-40}. Find two numbers that multiply to -40 and add to -3. These numbers are -8 and 5. Therefore, factor the numerator as follows: \( k^{2} - 3k - 40 = (k - 8)(k + 5)\).

Step 2: Factor the Denominator

The denominator of the fraction is \text{k^{2}-4k-45}. Find two numbers that multiply to -45 and add to -4. These numbers are -9 and 5. Therefore, factor the denominator as follows: \( k^{2} - 4k - 45 = (k - 9)(k + 5)\).

Step 3: Simplify the Fraction

Now, rewrite the fraction with the factored numerator and denominator:\( \frac{(k - 8)(k + 5)}{(k - 9)(k + 5)} \). Since \text{\( k + 5 \)} is common in both numerator and denominator, it can be canceled out, leaving:\( \frac{k - 8}{k - 9} \).

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra. It involves breaking down a polynomial into simpler components, called factors, that multiply together to give the original polynomial. This process is vital for simplifying algebraic fractions.
The first step is to identify pairs of numbers that multiply to the constant term and add to the coefficient of the linear term.
For example, let's factor the polynomial \(k^2 - 3k - 40\). We're looking for two numbers that multiply to \(-40\) and add to \(-3\). These numbers are \(-8\) and \(5\).
So, we can write \(k^2 - 3k - 40\) as \((k - 8)(k + 5)\).
This technique can simplify complex algebraic problems and is an essential skill in algebra.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form. This process can include factoring, canceling common factors, and combining like terms.
For the given problem, simplifying the expression \(\frac{k^2 - 3k - 40}{k^2 - 4k - 45}\) starts by factoring both the numerator and the denominator.
After factoring, we have \(\frac{(k - 8)(k + 5)}{(k - 9)(k + 5)}\).
The next step involves canceling out common factors in the numerator and the denominator.
Here, \( (k + 5) \) is a common factor and can be canceled out, resulting in the simplified expression \(\frac{k - 8}{k - 9}\).
Simplifying algebraic expressions makes them easier to work with and solves the problem more efficiently.

Common Factors

Identifying and canceling common factors is a critical step in simplifying algebraic fractions.
A common factor is a term that appears in both the numerator and the denominator of a fraction.
In the given problem, after factoring, the term \((k + 5)\) appears in both the numerator and the denominator: \(\frac{(k - 8)(k + 5)}{(k - 9)(k + 5)}\).
By canceling out \((k + 5)\), we simplify the fraction to \(\frac{k - 8}{k - 9}\).
Always be mindful of common factors as they are key to reducing fractions to their simplest form while maintaining the same value for the expression.