Problem 51 - Elementary Algebra

Question

For exercises 49-52, simplify. $$ \frac{k^{3}}{k^{2}+14 k+40}+\frac{64}{k^{2}+14 k+40} $$

Step-by-Step Solution

Answer

\( \frac{k^2 - 4k + 16}{k + 10} \)

Step 1 - Factor the Denominator

Factor the quadratic expression in the denominator: \( k^2 + 14k + 40 \). Look for two numbers that multiply to 40 and add up to 14. These numbers are 4 and 10. Hence, we factor the denominator as: \( k^2 + 14k + 40 = (k + 4)(k + 10) \).

Step 2 - Rewrite the Expression

Rewrite the given expression using the factored form of the denominator: \( \frac{k^3}{(k+4)(k+10)} + \frac{64}{(k+4)(k+10)} \).

Step 3 - Combine the Fractions

Since the denominators are the same, combine the fractions into a single fraction: \( \frac{k^3 + 64}{(k+4)(k+10)} \).

Step 4 - Simplify the Numerator

Notice that \( k^3 + 64 \) is a sum of cubes. Use the formula for the sum of cubes: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \) where \( a = k \) and \( b = 4 \). Thus, \( k^3 + 64 = (k + 4)(k^2 - 4k + 16) \).

Step 5 - Simplify the Complete Expression

Substitute the factored form of the numerator back into the expression: \( \frac{(k + 4)(k^2 - 4k + 16)}{(k + 4)(k + 10)} \). Cancel the common factor \( (k + 4) \) in the numerator and denominator: \( \frac{k^2 - 4k + 16}{k + 10} \).

Key Concepts

Factoring Quadratics

Factoring quadratics means breaking down a quadratic expression into simpler factors that, when multiplied together, give the original expression. This process is essential in simplifying algebraic expressions. For example, the expression \( k^2 + 14k + 40 \) is factored by finding two numbers that multiply to 40 and add to 14. In this case, the numbers are 4 and 10, so we factor as \( (k + 4)(k + 10) \). This makes it easier to handle complex fractions and simplify the whole expression.

Sum of Cubes

In algebra, recognizing patterns like the sum of cubes can simplify expressions drastically. The general formula for the sum of cubes is \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \). In our exercise, we use \( k^3 + 64 \), where \( k = a \) and \( 4 = b \). Applying the formula makes it \( (k + 4)(k^2 - 4k + 16) \). Recognizing such forms avoids tedious long-hand calculations and aids in better manipulation of the expression.

Combining Fractions

Combining fractions becomes straightforward when they have a common denominator. It's essential to factor the expressions so that the denominators match up. In our example, both fractions \( \frac{k^3}{(k+4)(k+10)} \) and \( \frac{64}{(k+4)(k+10)} \) have the same denominator \((k + 4)(k + 10)\). Thus, we can combine them into a single fraction: \( \frac{k^3 + 64}{(k+4)(k+10)} \). This reduces the complexity, making it easier to apply further simplification techniques.

Canceling Common Factors

Canceling common factors is a crucial final step in simplifying algebraic expressions. After factoring the numerator and the denominator, any common factors in them can be canceled out. From our combined fraction \( \frac{k^3 + 64}{(k+4)(k+10)} \), we factor the numerator to get \( \frac{(k + 4)(k^2 - 4k + 16)}{(k + 4)(k + 10)} \). Here, \( (k + 4) \) is a common factor present in both the numerator and the denominator. Canceling \( (k + 4) \) from both results in a simplified expression: \( \frac{k^2 - 4k + 16}{k + 10} \). This step is crucial for reaching the most reduced form of the expression.