Problem 37 - Elementary and Intermediate Algebra

Question

Simplify each rational expression. See Example 4 $$\frac{5 x^{2}-10 x}{x^{2}-4 x+4}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{5x}{x-2}\) with \(x \neq 2\).

Step 1: Factorize the Numerator

The given numerator is \(5x^2 - 10x\). Notice that both terms in the numerator have a common factor of \(5x\). Factoring \(5x\) out of the numerator gives:\[5x(x - 2)\]

Step 2: Factorize the Denominator

The given denominator is \(x^2 - 4x + 4\). Notice that the denominator is a perfect square trinomial. This can be rewritten as:\[(x - 2)^2\]

Step 3: Express as a Single Fraction

After factoring, the rational expression becomes:\[\frac{5x(x - 2)}{(x - 2)^2}\]

Step 4: Simplify the Fraction

The expression \((x - 2)\) is common in both the numerator and the denominator. We can cancel out one \((x - 2)\) from the numerator and one from the denominator, leaving:\[\frac{5x}{x - 2}\]

Step 5: State Any Restrictions

Since we canceled a factor \((x - 2)\) from the denominator, we must state the restriction that \(x eq 2\), to prevent division by zero.

Key Concepts

Factoring Polynomials

Factoring polynomials is like breaking down a number into its basic building blocks — called factors. When dealing with polynomials, the goal is to rewrite them as a product of simpler polynomial terms. This process helps us to simplify and solve expressions with ease. Let's look at an example: the polynomial \(5x^2 - 10x\).

• Both terms have the common factor of \(5x\).
• To factor it out: divide each term by \(5x\).
• This gives us: \(5x(x - 2)\).

By recognizing and factoring out the greatest common factor, we can simplify expressions step by step. Knowing these basic tricks can reduce even complex expressions into manageable pieces.

Perfect Square Trinomial

A perfect square trinomial is a special type of polynomial that results when a binomial is squared. It appears as three terms, typically in the form of \(a^2 \pm 2ab + b^2\). When working with a perfect square trinomial, it's customary to condense it into a squared binomial. Consider the denominator \(x^2 - 4x + 4\).

• Recognize the structure: \(x^2 - 4x + 4\) corresponds to \((x - 2)^2\).
• Each part of the trinomial represents a term in the squared binomial.
• This makes calculations easier by simplifying it into a compact expression.

Spotting and rewriting perfect square trinomials allows the polynomial expressions to be simplified dramatically, not unlike solving a puzzle by finding matching pieces.

Fraction Simplification

Simplifying fractions involves reducing the expression to its simplest form. In rational expressions, the goal is to cancel matching factors in the numerator and denominator. For the expression \(\frac{5x(x - 2)}{(x - 2)^2}\):

• Both the numerator and denominator contain the factor \((x - 2)\).
• Cancel one \((x - 2)\) from both the top and bottom.
• Remaining expression: \(\frac{5x}{x - 2}\).

By canceling out the common factors, the fraction becomes simplified. This technique reduces complexity and helps make further calculations simpler to handle.

Division by Zero Restriction

When simplifying rational expressions, it's critical to consider division by zero. Division by zero is undefined in mathematics and can lead to incorrect answers. Thus, stating restrictions is an important practice to ensure mathematical accuracy.

• The original expression \(\frac{5x(x - 2)}{(x - 2)^2}\) loses a factor from the denominator.
• After simplification, we find: \(x eq 2\).
• This restriction comes from the original denominator \((x - 2)^2\), which indicates \(x\) cannot be \(2\), else it results in division by zero.

Being mindful of these restrictions maintains the integrity of the solution and prevents errors in mathematical operations.