Question

Find the product. $$ \frac{x^2-4 x}{x-1} \cdot \frac{x^2+3 x-4}{2 x} $$

Short answer

The result of the specified multiplication is \(\frac{(x - 4)(x + 4)}{2}\).

Step-by-step solution

Factorize the Numerators and Denominators

First factorize the numerator \(x^2 - 4x\) of the first fraction and the numerator \(x^2 + 3x - 4\) of the second fraction. The denominators \(x - 1\) and \(2x\) are already factorized. So, the expression becomes \[\frac{x(x - 4)}{x - 1} \cdot \frac{(x - 1)(x + 4)}{2x}\].

Cancel out Common Factors

Cancel out the same factors appearing both in the numerator and denominator. Here, you can cancel out \(x - 1\) on both numerator and denominator and \(x\) can be canceled in the second fraction and the first fraction. Hence, the expression becomes \[\frac{(x - 4)}{1} \cdot \frac{(x + 4)}{2}\].

Multiply Simplified Fractions

Now multiply the simplified fractions, where the numerator of the first fraction multiplies the numerator of the second fraction and the denominator of the first fraction multiplies the denominator of the second fraction. The final result becomes \[\frac{(x - 4)(x + 4)}{2}\].

Key concepts

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler terms or factors that, when multiplied together, give back the original polynomial. This process is helpful when working with algebraic fractions. For example, consider the polynomial \( x^2 - 4x \).

• First, we look for the greatest common factor (GCF) of the terms in the polynomial. In this case, both terms contain \( x \), making \( x \) the GCF.
• Next, factor out \( x \): \( x(x - 4) \).

Similarly, the quadratic polynomial \( x^2 + 3x - 4 \) can be factored by finding two numbers that multiply to \(-4\) (the constant term) and add up to \(3\) (the linear coefficient). These numbers are \(4\) and \(-1\), so the polynomial factors as \((x - 1)(x + 4)\).
Factoring simplifies expressions and is crucial for further manipulation and simplification of algebraic fractions.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Like fractions, rational expressions can be added, subtracted, multiplied, or divided.
It's important to factor both the numerators and the denominators of rational expressions to identify common factors that can be canceled.
In our example:

• The first fraction is \(\frac{x(x - 4)}{x - 1}\).
• The second fraction is \(\frac{(x - 1)(x + 4)}{2x}\).

By factoring, you easily view and eliminate common factors, simplifying the rational expression.
It's crucial to remember that the expressions remain unchanged in value when equivalent fractions are formed through cancellation of common factors.

Simplifying Fractions

Simplifying fractions is the process of reducing fractions to their simplest form. This is done by canceling out factors that appear in both the numerator and the denominator.
In the expression \(\frac{x(x - 4)}{x - 1} \cdot \frac{(x - 1)(x + 4)}{2x}\), you should:

• Identify and cancel the common factor \( x - 1 \) present in both the numerator and denominator.
• Cancel the factor \( x \) from the second fraction and numerator of the first fraction.

The remaining expression \(\frac{(x - 4)(x + 4)}{2}\) is the simplified form.
Simplifying fractions is key to ease calculations, providing clean and manageable expressions, and often clarifies the functions and computations involved.