Questions: Perform the following operation and express in simplest form. 4x / (x^2 + x - 42) ÷ x^3 / (x^2 - 49)

Solution Steps

We start by factorizing the polynomials involved in the expression:

1. The polynomial \(x^{2}+x-42\) factors to \((x - 6)(x + 7)\).
2. The polynomial \(x^{3}\) remains as \(x^{3}\).
3. The polynomial \(x^{2}-49\) factors to \((x - 7)(x + 7)\).

Thus, we have:

• \(x^{2}+x-42 = (x - 6)(x + 7)\)
• \(x^{3} = x^{3}\)
• \(x^{2}-49 = (x - 7)(x + 7)\)

The original expression can be rewritten as follows: \[ \frac{4 x}{x^{2}+x-42} \div \frac{x^{3}}{x^{2}-49} = \frac{4 x}{(x - 6)(x + 7)} \times \frac{(x - 7)(x + 7)}{x^{3}} \]

Now, we multiply the two rational expressions: \[ \frac{4 x (x - 7)(x + 7)}{x^{3} (x - 6)(x + 7)} \]

Next, we can cancel the common factor \((x + 7)\) from the numerator and denominator: \[ \frac{4 x (x - 7)}{x^{3} (x - 6)} \]

This simplifies to: \[ \frac{4 (x - 7)}{x^{2} (x - 6)} \]

Final Answer