Problem 44 - Algebra 1

Question

Which of the following represents the expression \(\frac{x^{2}-3 x}{x^{2}-5 x+6} \cdot \frac{(x-2)^{2}}{2 x}\) in simplified form? (A) \(\frac{x(x-3)}{2}\) (B) \(\frac{x}{2}\) (C) \(\frac{x-2}{2}\) (D) \(\frac{x(x-3)}{x-2}\) (E) \(\frac{x^{2}-4 x+4}{x-2}\)

Step-by-Step Solution

Answer

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

Step 1: Factorize the Expressions

Firstly, factorize the expressions in the numerator and the denominator into irreducible factors if possible. We will start with \(\frac{x^{2}-3 x}{x^{2}-5 x+6}\). For that, the numerator simplifies to: \(x(x-3)\) and the denominator to: \((x-3)(x-2)\).\nThe whole expression now looks like this: \(\frac{x(x-3)}{(x-3)(x-2)}.\)

Step 2: Simplify and Rewrite the Expression

We can cancel \(x-3\) from the numerator and the denominator, and this gives us a simplified form as: \(\frac{x}{x-2}\). Now, rewrite the entire expression by multiplying this with \(\frac{(x-2)^{2}}{2x}\) which gives \(\frac{x}{x-2} \cdot \frac{(x-2)^{2}}{2x}\).

Step 3: Further Simplification

We can now cancel \(x\) and \(x-2\) to yield the final simplified expression as: \(\frac{x-2}{2}\) or \(0.5(x-2)\)

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial process in algebra that involves breaking down complex expressions into products of simpler ones. The purpose is to find the most basic expressions, called factors, that when multiplied together give back the original polynomial. This operation is particularly helpful when simplifying algebraic fractions, solving polynomial equations, or finding zeros of polynomial functions.
For instance, a quadratic polynomial like x^2 - 5x + 6 can be factored by finding two numbers that multiply to give the constant term, 6, and also add up to give the coefficient of the x term, which is -5. These numbers are -3 and -2, leading to the factorization (x - 3)(x - 2). Recognizing these factors enables us to simplify expressions easily as it did in the given exercise.

• Check for common factors in the terms of the polynomial.
• Use techniques such as the difference of squares, sum/product of cubes, or the quadratic formula, if applicable.
• For higher-degree polynomials, look for patterns or use synthetic division or the factor theorem.

It's important to practice factoring with various types of polynomials to become proficient and to make subsequent algebraic operations significantly easier.

Simplifying Expressions

Simplifying expressions means to rewrite them in their most basic form without changing their values. This process typically involves combining like terms, reducing fractions, or eliminating common factors.
In the context of our example, after factoring polynomials, we notice that x(x-3) and (x-3)(x-2) share a common factor, x-3. By canceling this common factor, we simplify the expression to x/(x-2), making the expression less complex. When simplifying:

• Always look for and cancel out common factors if possible.
• Combine like terms and use associative and commutative properties.
• Remember that simplification does not mean approximation; the expressions value does not change.

Simplification can also involve rationalizing denominators, expanding expressions using distributive property, and factoring when necessary, as illustrated in the given problem.

Rational Expressions

Rational expressions are fractions that contain polynomials in both their numerators and denominators. Simplifying rational expressions is similar to simplifying numerical fractions by canceling common factors. However, with rational expressions, those factors are often polynomials or binomials that require factoring beforehand, as shown in our exercise.
In simplifying rational expressions:

• Factor both the numerator and the denominator to identify and cancel common factors.
• Look out for expressions that can be simplified by basic algebraic operations, like distributing or combining like terms.
• Pay special attention to restrictions, such as values that make the denominator zero, which are not allowed.

In the exercise, once the common factors are canceled out after factoring, we arrived at x/(x-2). When multiplied with another rational expression (x-2)^2/(2x), further cancellation happens, leading to the final simplified rational expression, (x-2)/2. Understanding rational expressions and their simplification is fundamental to algebra and higher mathematics.