Problem 84 - Intermediate Algebra

Question

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{x^{3}+8}{x^{4}-2 x^{3}+4 x^{2}} $$

Step-by-Step Solution

Answer

\( \frac{x+2}{x^2} \)

Step 1: Identify Common Factors

First, look at the numerator and denominator to see if there are any common factors that might allow us to simplify the expression. The expression is \( \frac{x^{3}+8}{x^{4}-2x^{3}+4x^{2}} \). Notice the numerator \( x^3 + 8 \) can be seen as a sum of cubes: \( x^3 + 2^3 \). The denominator doesn't immediately simplify by factoring.

Step 2: Factor the Numerator

Factor the numerator using the sum of cubes formula, which states \( a^3 + b^3 = (a+b)(a^2-ab+b^2) \) where \( a = x \) and \( b = 2 \). Therefore, \( x^3 + 8 = (x+2)(x^2-2x+4) \).

Step 3: Factor the Denominator

Factor out the greatest common factor in the denominator. The expression \( x^4 - 2x^3 + 4x^2 \) has a common factor of \( x^2 \), so factor it as \( x^2(x^2 - 2x + 4) \).

Step 4: Simplify the Expression

Now that we have factored both the numerator and the denominator, substitute these back into the original expression: \[ \frac{(x+2)(x^2-2x+4)}{x^2(x^2-2x+4)} \]Cancel the common factor \( x^2 - 2x + 4 \) from both the numerator and the denominator since it appears in both.

Step 5: Rewrite the Simplified Expression

After canceling the common factor, the expression simplifies to \[ \frac{x+2}{x^2} \].

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra involves reducing them to their simplest form. This makes mathematical expressions easier to work with and understand. To simplify an expression, we aim to remove unnecessary complexity without changing its value.

In algebraic fractions, simplification often involves both the numerator and the denominator. It is important to identify and cancel out any common factors or equivalent expressions present in both parts of the fraction. This is particularly useful in making an equation more manageable.

Here are general steps to simplify an algebraic fraction:

• Identify common factors in the numerator and the denominator.
• Factorize both numerator and denominator completely.
• Cancel out the common factors that appear in both the numerator and the denominator.
• Simplify the resulting expression as much as possible.

Simplifying not only streamlines the expression but also helps in solving equations more effectively. This process is crucial in managing complex algebraic tasks.

Factoring Techniques

Factoring is a key technique in algebra for simplifying expressions and solving equations. It involves expressing an algebraic expression as a product of simpler factors.

Here are a few essential factoring techniques:

• **Greatest Common Factor (GCF):** Identify and factor out the largest expression or number that divides all terms of the polynomial.
• **Difference of Squares:** Recognize expressions in the form of \( a^2 - b^2 \), which can be factored as \( (a+b)(a-b) \).
• **Trinomials:** Factor trinomials into the product of two binomials. For example, \( x^2 + bx + c \) can often be factored as \( (x+m)(x+n) \).
• **Sum and Difference of Cubes:** Use the formulas \( a^3 + b^3 = (a+b)(a^2-ab+b^2) \) for sums and \( a^3 - b^3 = (a-b)(a^2+ab+b^2) \) for differences.

Each of these techniques has its own set of rules and formats. Mastering these allows you to tackle a variety of algebraic challenges, particularly in simplifying expressions, solving equations, and finding roots of polynomials.

Sum of Cubes Formula

In algebra, the sum of cubes formula is a specific method used to factor expressions of the form \( a^3 + b^3 \). This formula is particularly helpful in simplifying complex expressions.

The sum of cubes formula is written as:
\[ a^3 + b^3 = (a+b)(a^2 - ab + b^2) \]
This formula decomposes a sum of cubes into a product of a binomial and a trinomial. In this format:

• \( a \) and \( b \) are any two expressions that are cubed.
• The trinomial \( a^2 - ab + b^2 \) cannot be factored further and ensures the factorization is complete.

In practical use, the sum of cubes formula allows us to factorize and simplify expressions that might otherwise be difficult to manipulate. Understanding this formula is crucial when dealing with polynomial expressions involving cubes, as seen in simplifying algebraic fractions.