Problem 68 - Elementary Algebra

Question

$$ \text { For exercises 67-72, simplify. } $$ $$ \frac{x^{3}+27}{x^{2}-9} $$

Step-by-Step Solution

Answer

\(\frac{x^2 - 3x + 9}{x - 3}\)

Step 1 - Factorize the Numerator

The numerator is a sum of cubes: \[x^3 + 27 = x^3 + 3^3\] Utilize the sum of cubes formula: \[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\] In this case, \(a = x\) and \(b = 3\). Applying the formula: \[x^3 + 27 = (x + 3)(x^2 - 3x + 9)\]

Step 2 - Factorize the Denominator

The denominator is a difference of squares: \[x^2 - 9 = x^2 - 3^2\] Use the difference of squares formula: \[a^2 - b^2 = (a - b)(a + b)\] Here, \(a = x\) and \(b = 3\). Applying the formula: \[x^2 - 9 = (x - 3)(x + 3)\]

Step 3 - Simplify

Write the expression with the factored forms of the numerator and denominator: \[\frac{(x+3)(x^2 - 3x + 9)}{(x-3)(x+3)}\] Cancel the common term \((x + 3)\): \[\frac{(x^2 - 3x + 9)}{(x - 3)}\]

Key Concepts

Sum of Cubes

Algebra often involves working with different types of polynomials. One special case is the **sum of cubes**. The formula for the sum of cubes is given by: oindent\[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]
To utilize this formula, identify if the expression matches the form \(a^3 + b^3\). For example, in the expression \(x^3 + 27\), we can see that it matches \(x^3 + 3^3\) with \(a = x\) and \(b = 3\). Using the sum of cubes formula, we get:oindent\[x^3 + 27 = (x + 3)(x^2 - 3x + 9)\]
This method helps in breaking down complex polynomial expressions into simpler factors.

Difference of Squares

Another common polynomial pattern is the **difference of squares**. This pattern follows the formula:oindent\[a^2 - b^2 = (a - b)(a + b)\]
When you see an expression like \(x^2 - 9\), it can be factored because it matches the form \(x^2 - 3^2\). Setting \(a = x\) and \(b = 3\), we apply the formula and get:oindent\[x^2 - 9 = (x - 3)(x + 3)\]
Recognizing and applying this pattern simplifies the expression quickly. This is highly useful in algebraic manipulations, particularly when simplifying complex fractions.

Factoring Polynomials

Factoring polynomials is a key skill in algebra. It involves writing a polynomial as a product of its factors. For example, consider the numerator and denominator of the fraction given by:oindent\[\frac{x^3 + 27}{x^2 - 9}\]**Step 1: Factoring the Numerator**: We identified earlier that \(x^3 + 27\) is a sum of cubes, so we factorize it using the sum of cubes formula:oindent\[x^3 + 27 = (x + 3)(x^2 - 3x + 9)\]**Step 2: Factoring the Denominator**: The denominator \(x^2 - 9\) is a difference of squares. Using the difference of squares formula, we get:oindent\[x^2 - 9 = (x - 3)(x + 3)\]**Step 3: Simplifying the Expression**: Now, we write the fraction with the factored forms:oindent\[\frac{(x + 3)(x^2 - 3x + 9)}{(x - 3)(x + 3)}\] Cancel out the common factor \((x + 3)\) from the numerator and denominator to simplify the fraction:oindent\[\frac{x^2 - 3x + 9}{x - 3}\]
Factoring polynomials makes it easier to perform operations such as addition, subtraction, and simplification.