Question

1. (a) simplify: \frac{x^3+8x^2+15x}{x^2-x-2} \div \frac{x^2+5x}{x^2-4}

Explanation:

⚡ Using: factoring and solving quadratic equations · 🆕 New: Simplifying Rational Expressions
Dividing fractions means multiplying by the reciprocal

Step 1: Convert division to multiplication

To divide by a rational expression, multiply by its reciprocal (flip the second fraction):

$$ \frac{x^3 + 8x^2 + 15x}{x^2 - x - 2} \cdot \frac{x^2 - 4}{x^2 + 5x} $$

Step 2: Factor all numerators and denominators

Factor each polynomial completely:

1. First numerator: \(x^3 + 8x^2 + 15x\)

• Factor out the greatest common factor, \(x\):

$$ x(x^2 + 8x + 15) $$

• Factor the quadratic inside: find two numbers that multiply to \(15\) and add to \(8\) (\(3\) and \(5\)):

$$ x(x + 3)(x + 5) $$

1. First denominator: \(x^2 - x - 2\)

• Find two numbers that multiply to \(-2\) and add to \(-1\) (\(-2\) and \(1\)):

$$ (x - 2)(x + 1) $$

1. Second numerator: \(x^2 - 4\)

• This is a difference of squares, \(a^2 - b^2 = (a - b)(a + b)\):

$$ (x - 2)(x + 2) $$

1. Second denominator: \(x^2 + 5x\)

• Factor out the greatest common factor, \(x\):

$$ x(x + 5) $$

Step 3: Combine and simplify

Substitute the factored forms back into the expression:

$$ \frac{x(x + 3)(x + 5)}{(x - 2)(x + 1)} \cdot \frac{(x - 2)(x + 2)}{x(x + 5)} $$

Combine into a single fraction:

$$ \frac{x(x + 3)(x + 5)(x - 2)(x + 2)}{(x - 2)(x + 1)x(x + 5)} $$

Cancel out the common factors that appear in both the numerator and the denominator (\(x\), \(x + 5\), and \(x - 2\)):

$$ \frac{\cancel{x}(x + 3)\cancel{(x + 5)}\cancel{(x - 2)}(x + 2)}{\cancel{(x - 2)}(x + 1)\cancel{x}\cancel{(x + 5)}} = \frac{(x + 3)(x + 2)}{x + 1} $$

Expand the numerator:

$$ (x + 3)(x + 2) = x^2 + 5x + 6 $$

So, the simplified expression is:

$$ \frac{x^2 + 5x + 6}{x + 1} $$

Answer:

$$ \frac{(x + 3)(x + 2)}{x + 1} \quad \text{or} \quad \frac{x^2 + 5x + 6}{x + 1} $$