Problem 72 - Elementary Algebra

Question

$$ \text { For exercises 67-72, simplify. } $$ $$ \frac{k+5}{k^{3}+125} $$

Step-by-Step Solution

Answer

$$\frac{1}{k^2 - 5k + 25}$$

Step 1: Recognize the form of the denominator

Notice that the denominator is in the format of a sum of cubes. The expression can be written as $$ k^3 + 125 = k^3 + 5^3 $$

Step 2: Factor the sum of cubes

The sum of cubes can be factored using the formula: $$ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $$In this case, let \(a = k\) and \(b = 5\). Thus, $$ k^3 + 5^3 = (k + 5)(k^2 - 5k + 25) $$

Step 3: Substitute the factored form into the original expression

Replace the denominator with its factored form: $$ \frac{k+5}{k^3+125} = \frac{k+5}{(k+5)(k^2 - 5k + 25)} $$

Step 4: Simplify the expression

We can now cancel the common factor, \((k+5)\), in the numerator and the denominator: $$ \frac{k+5}{(k+5)(k^2 - 5k + 25)} = \frac{1}{k^2 - 5k + 25} $$

Key Concepts

algebraic factoring

One of the main tools used in solving algebraic equations is factoring. Factoring is the process of breaking down an expression into simpler terms, or 'factors', that can be multiplied together to obtain the original expression. When we encounter complex polynomials, factoring can simplify the problem by reducing it to more manageable parts.
For instance, in the exercise, recognizing the denominator as a sum of cubes allows us to apply specific factoring formulas. Factoring is not just limited to cubes; it also includes techniques like extracting common factors, factoring trinomials, and using difference of squares.
Understanding how to factor various types of expressions is crucial in algebra as it simplifies solving and simplifies equations significantly.

simplifying rational expressions

Simplifying rational expressions involves reducing a fraction where both the numerator and the denominator are polynomials. The goal is to express it in its simplest form. In the provided solution, after factoring the denominator, we noticed a common factor of \(k+5\) in both the numerator and the denominator.
By canceling out this common factor, we streamlined the expression. This process requires a good grasp of factorization and algebraic identities.
It's essential to look out for these common factors to simplify rational expressions quickly. Always ensure the final form is as simple as possible, which helps in further manipulations and solutions.

sum of cubes formula

The sum of cubes formula is a special factoring formula that applies to expressions of the form \(a^3 + b^3\). This formula states:
\[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]
It's crucial for algebraic problems where you'll regularly encounter polynomials that fit this pattern. In our exercise, recognizing that \(k^3 + 125\) can be written as \(k^3 + 5^3\) was key to simplifying the expression.
By setting \(a = k\) and \(b = 5\), we applied the sum of cubes formula to transform the expression into a more workable form.
Knowing this formula allows you to break down specific types of polynomials efficiently, facilitating easier manipulation and simplification.

algebraic expressions

Algebraic expressions are combinations of variables, constants, and operators (like addition, subtraction, multiplication, and division). Understanding how to work with these expressions is the foundation of algebra.
Manipulating algebraic expressions involves operations such as simplification, factoring, and substitution. The goal is often to simplify complex expressions or solve equations.
In solving the given exercise, recognizing the structure of the expression as a sum of cubes and factoring it correctly allowed us to simplify it significantly. Mastery of these basic manipulations is essential for progressing in algebra and tackling more complex problems.
Always look for patterns and structures within the expression that can lead to simpler forms. This approach makes solving algebraic problems more manageable and intuitive.