Problem 43 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\( \frac{5x}{x - 2} \)

Step 1: Factor the Numerator

The numerator is \(5x^2 - 10x\). We can factor out the greatest common factor, which is \(5x\). This gives us: \[5x(x - 2)\].

Step 2: Factor the Denominator

The denominator is \(x^2 - 4x + 4\). This expression is a perfect square trinomial and can be factored as: \[(x - 2)^2\].

Step 3: Simplify the Expression

Now, substitute the factored forms back into the original expression: \[\frac{5x(x - 2)}{(x - 2)^2}\]. We can simplify by canceling the common factor of \((x - 2)\) in the numerator and the denominator, which results in: \[\frac{5x}{x - 2}\]. Note that \(x eq 2\) to avoid dividing by zero.

Key Concepts

Factoring Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients, built through addition, subtraction, and multiplication. When you "factor" a polynomial, you're essentially breaking it down into simpler, multiplied components, much like reducing a fraction to its smallest terms.

This technique makes solving polynomials straightforward, particularly when simplifying rational expressions. Let's consider the given exercise. The polynomial in the numerator is \(5x^2 - 10x\). Here's how to factor it:

• Identify the greatest common factor (GCF). In this case, it's \(5x\) because both terms share this factor.
• Extract the GCF, and rewrite the polynomial as \(5x(x - 2)\).

By doing this, we've expressed the original polynomial as a product of its factors, making it more manageable, especially when combined expressions or equations are involved.

Perfect Square Trinomial

A perfect square trinomial is a special kind of polynomial that can be expressed as the square of a binomial. Recognizing these can simplify factoring immensely. The denominator in the provided exercise, \(x^2 - 4x + 4\), fits this description.

To determine whether a trinomial is a perfect square, check if it matches the pattern \((a - b)^2 = a^2 - 2ab + b^2\). Here's how it applies:

• Identify \(a\) and \(b\) such that \(a^2 = x^2\) and \(b^2 = 4\). Thus, \(a = x\) and \(b = 2\).
• Verify that \(-2ab\) matches the middle term \(-4x\), which indeed it does, as \(-2(x)(2) = -4x\).
• Hence, \(x^2 - 4x + 4 = (x - 2)^2\).

Recognizing and factoring out perfect square trinomials simplifies expressions and aids in canceling out terms during division.

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest factor that two or more numbers or terms share. Emphasizing the use of the GCF can be extremely useful when simplifying fractions or expressions, especially in polynomial factoring.
In our example, the numerator \(5x^2 - 10x\) is made simpler through the GCF method:

• First, identify the common factors of each term: the factors of \(5x^2\) are \(1, 5, x, x^2, 5x, 5x^2\) and of \(10x\) are \(1, 2, 5, 10, x, 2x, 5x\).
• By comparing, \(5x\) is identified as the largest shared factor.
• Factor \(5x\) out, rewriting \(5x^2 - 10x\) as \(5x(x - 2)\).

Using the GCF simplifies polynomial problems by reducing complex expressions into basic components that are easier to evaluate or cancel.