Problem 67 - Elementary Algebra

Question

$$ \text { For exercises 67-72, simplify. } $$ $$ \frac{x^{3}+8}{x^{2}-4} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{x^2 - 2x + 4}{x - 2} \).

Step 1: Identify the given expression

The given expression to simplify is \( \frac{x^{3}+8}{x^{2}-4} \).

Step 2: Factor the numerator

Recognize that the numerator \( x^3 + 8 \) is a sum of cubes. Use the formula \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \) where \( a = x \) and \( b = 2 \). Thus, \( x^3 + 8 = (x + 2)(x^2 - 2x + 4) \).

Step 3: Factor the denominator

Recognize that the denominator \( x^2 - 4 \) is a difference of squares. Use the formula \( a^2 - b^2 = (a - b)(a + b) \) where \( a = x \) and \( b = 2 \). Thus, \( x^2 - 4 = (x - 2)(x + 2) \).

Step 4: Substitute the factors back into the fraction

Substitute the factored forms into the original expression: \( \frac{x^{3}+8}{x^{2}-4} = \frac{(x + 2)(x^2 - 2x + 4)}{(x - 2)(x + 2)} \).

Step 5: Simplify the expression

Cancel the common factor \( x + 2 \) from the numerator and the denominator: \( \frac{(x + 2)(x^2 - 2x + 4)}{(x - 2)(x + 2)} = \frac{x^2 - 2x + 4}{x - 2} \).

Step 6: Present the simplified expression

The simplified form of the original expression is \( \frac{x^2 - 2x + 4}{x - 2} \).

Key Concepts

Sum of Cubes

When given the sum of cubes algebraic expression, such as \(a^3 + b^3\), we use a specific formula to break it down into a more manageable form. The formula for the sum of cubes is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
Let's apply this to our numerator \(x^3 + 8\). Here, \(a = x\) and \(b = 2\). Using the sum of cubes formula, we get:
\[ x^3 + 8 = (x + 2)(x^2 - 2x + 4) \]
This factorization helps to simplify the expression by breaking it into two simpler polynomial factors. This step is essential before we move on to handle the denominator for simplification.

Difference of Squares

For the denominator, we notice that we have a difference of squares, which appears in the form of \(a^2 - b^2\). The formula to factor a difference of squares is \((a - b)(a + b)\). This is a crucial pattern to recognize in algebraic fractions.
In our problem, \(x^2 - 4\) fits the difference of squares pattern where \(a = x\) and \(b = 2\). Applying the difference of squares formula, we get:
\[ x^2 - 4 = (x - 2)(x + 2) \]
With the denominator factored, it becomes easier to see common factors in the fraction, setting us up to simplify it further.

Factoring

Factoring involves breaking down an expression into simpler terms that, when multiplied together, give back the original expression. It's a key step in simplifying fractions and solving equations.
In our exercise, after factoring both the numerator and the denominator, we have:
\[ \frac{(x + 2)(x^2 - 2x + 4)}{(x - 2)(x + 2)} \]
Notice that \((x + 2)\) is a factor both in the numerator and the denominator. This common factor can be canceled out, which simplifies our expression:
\[ \frac{(x + 2)(x^2 - 2x + 4)}{(x - 2)(x + 2)} = \frac{x^2 - 2x + 4}{x - 2} \]
Canceling out common factors is a fundamental step in simplifying algebraic fractions. Make sure to check that the term you're canceling is not equal to zero, to avoid undefined expressions.