Problem 83 - Intermediate Algebra

Question

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{x^{4}+3 x^{3}+9 x^{2}}{x^{3}-27} $$

Step-by-Step Solution

Answer

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

Step 1: Factor the numerator

Look at the numerator \(x^4 + 3x^3 + 9x^2\). Notice that each term has a common factor of \(x^2\). Factor out \(x^2\) from each term:\[x^4 + 3x^3 + 9x^2 = x^2(x^2 + 3x + 9)\]

Step 2: Factor the denominator

Look at the denominator \(x^3 - 27\). This expression is a difference of cubes, which can be factored using the formula \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Let \(a = x\) and \(b = 3\):\[x^3 - 27 = (x - 3)(x^2 + 3x + 9)\]

Step 3: Simplify the expression

Now that both the numerator and the denominator have been factored, rewrite the original expression using these factored forms:\[\frac{x^2(x^2 + 3x + 9)}{(x-3)(x^2 + 3x + 9)}\]Notice that \(x^2 + 3x + 9\) appears in both the numerator and the denominator. Cancel these common factors:\[\frac{x^2 \cancel{(x^2 + 3x + 9)}}{(x-3)\cancel{(x^2 + 3x + 9)}} = \frac{x^2}{x-3}\]

Step 4: Write the simplified expression

After canceling the common factor, the expression simplifies to:\[\frac{x^2}{x-3}\] This is the simplified form of the original expression.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra, helping to simplify expressions, solve equations, and uncover important characteristics of a mathematical expression. When we factor a polynomial, we essentially rewrite it as a product of simpler expressions, called factors. This process is somewhat like reversing the distributive property, spreading out the components of the polynomial into multiplied groups.

For instance, in the original exercise, the numerator was given as \(x^4 + 3x^3 + 9x^2\). Recognizing that each term shares a common factor, we factor out the smallest shared power of a variable, in this case, \(x^2\). This gives us:

• A single simplified factor: \(x^2(x^2 + 3x + 9)\)

Breaking each term into these factors makes future simplification or division steps more straightforward.

Simplifying Expressions

Simplifying algebraic expressions involves reducing them to their most concise forms while holding on to their inherent value. This often means factoring polynomials and canceling out any repeated terms or factors in fractions. The overarching goal is achieving a simpler, more streamlined expression.

In our exercise, after factoring both the numerator and the denominator, we noticed the term \(x^2 + 3x + 9\) appears in both places:

• Numerator: \(x^2(x^2 + 3x + 9)\)
• Denominator: \((x-3)(x^2 + 3x + 9)\)

The common term \((x^2 + 3x + 9)\), acts like a mutual factor that can be canceled out, as it essentially divides by itself to become 1. After this cancellation step, we arrive at the simplified expression \(\frac{x^2}{x-3}\). This process of elimination ensures our final answer is both precise and comprehensible.

Difference of Cubes

The difference of cubes is a specific algebraic formula used to factor expressions in the form of \(a^3 - b^3\). This method is pivotal in simplifying expressions that otherwise appear complex. The formula states:

• \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)

Recognizing this pattern can help us factor such expressions correctly.

In our exercise context, the denominator \(x^3 - 27\) is a classic difference of cubes, where \(a = x\) and \(b = 3\). Plugging into the formula, we get:

• \(x^3 - 27 = (x - 3)(x^2 + 3x + 9)\)

This factorization is key, as it exposes a common factor with the numerator, which facilitates the simplification of the entire expression.