Question

Simplify \(\frac{x^2-5 x}{4 x^2} \cdot \frac{x^2-7 x+12}{x^2-16} \div \frac{2 x-10}{x^2+2 x-8}\). Write your answer in lowest terms. A. \(\frac{(x+3)(x+2)}{8 x}\) B. \(\frac{(x-3)(x+2)}{8 x}\) C. \(\frac{(x+3)(x-2)}{8 x}\) D. \(\frac{(x-3)(x-2)}{8 x}\)

Short answer

\(\frac{(x-3)(x-2)}{8x}\)

Step-by-step solution

Factorize the Numerator and Denominator

Identify and factorize the polynomials in each fraction. For \(\frac{x^2 - 5x}{4x^2}\): numerator can be written as \(x(x-5)\) and the denominator remains \(4x^2\). For \(\frac{x^2-7x+12}{x^2-16}\): numerator can be written as \( (x-3)(x-4)\) and denominator as \( (x-4)(x+4)\). For \(\frac{2x-10}{x^2+2x-8}\): numerator as \( 2(x-5)\) and the denominator is \( (x-2)(x+4)\).

Combine and Simplify Fractions

Rewrite the original expression using the factored forms: \[ \left( \frac{x(x-5)}{4x^2} \right) \cdot \left( \frac{(x-3)(x-4)}{(x-4)(x+4)} \right) \div \left( \frac{2(x-5)}{(x-2)(x+4)} \right) \]

Dividing by a Fraction

Change the division to multiplication by taking the reciprocal of the third fraction: \[ \left( \frac{x(x-5)}{4x^2} \right) \cdot \left( \frac{(x-3)(x-4)}{(x-4)(x+4)} \right) \cdot \left( \frac{(x-2)(x+4)}{2(x-5)} \right) \]

Cancel Common Factors

Cancel out the common factors in the numerator and the denominator: \[ x^2: \left( \frac{\cancel{x}(x-5)}{4x^2} \right) \cdot \left( \frac{(x-3)\cancel{(x-4)}}{\cancel{(x-4)}(x+4)} \right) \cdot \left( \frac{(x-2)\cancel{(x+4)}}{2\cancel{(x-5)}} \]\1x\ cancels with the numerator x and 4 remains. }

Simplify the Expression

Multiply the remaining factors: \[ \frac{(x-3)(x-2)}{8x} \]

Key concepts

Factoring Polynomials

Factoring polynomials is a crucial step in simplifying algebraic expressions. Essentially, you break down a polynomial into products of its factors. Think of it as reverse distributing. For instance, consider the polynomial \( x^2 - 5x \). You notice that \( x \) is common in both terms. So, you factor it out: \( x^2 - 5x = x(x - 5) \). This method helps simplify further calculations by making multiplication and division easier.
When you encounter more complex polynomials, such as \( x^2 - 7x + 12 \), you look for two numbers that multiply to give the constant term (here, 12) and add to give the middle coefficient (here, -7). Thus, the factors are \( x - 3 \) and \( x - 4 \), making the polynomial \( (x - 3)(x - 4) \).
Factoring is the backbone of simplifying algebraic expressions, so practice it thoroughly.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing fractions by canceling common factors from the numerator and the denominator. Just as you would simplify a regular fraction like \( \frac{4}{8} \to \frac{1}{2} \), you follow similar steps with algebraic fractions.
Take the fraction \( \frac{x(x - 5)}{4x^2} \). You notice that \( x \) appears in both the numerator and the denominator. Thus, you can cancel the \( x \) leaving you with \( \frac{x - 5}{4x} \).
Next, consider combining fractions. Write them in factored form to simplify further: \( \frac{(x - 3)(x - 4)}{(x - 4)(x + 4)} \times \frac{(x - 2)(x + 4)}{2(x - 5)} \). Cancel out common factors: \( x - 4 \) from numerator and denominator. Now, multiply the remaining terms, paying attention to simplify wherever possible. This results in a new simplified fraction.
The key to simplifying algebraic fractions is to methodically cancel out common factors and reduce step by step.

Multiplication and Division of Fractions

When dealing with multiplication and division of fractions in algebra, the steps are straightforward but need careful attention to detail. For multiplication, simply multiply the numerators together and the denominators together. For instance, \( \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \).
In the case of dividing fractions, change the division sign to multiplication and take the reciprocal of the fraction after the division sign. For example, \( \frac{a}{b} \times \frac{d}{c} \). Now multiply as usual.
In our exercise, we initially have a division: \( \frac{x(x - 5)}{4x^2} \times \frac{(x - 3)(x - 4)}{(x - 4)(x + 4)} \times \frac{(x - 2)(x + 4)}{2(x - 5)} \). Rewriting the division as multiplication: \( \frac{x(x - 5)}{4x^2} \times \frac{(x - 3)(x - 4)}{(x - 4)(x + 4)} \times \frac{(x - 2)(x + 4)}{2(x - 5)} \). Now cancel and simplify step-by-step.
Multiplying and dividing fractions becomes simpler with practice and attention to detail involved in each step.