Problem 34 - Algebra and Trigonometry

Question

Perform the multiplication or division and simplify. $$ \frac{x^{2}-x-6}{x^{2}+2 x} \cdot \frac{x^{3}+x^{2}}{x^{2}-2 x-3} $$

Step-by-Step Solution

Answer

The simplified expression is \(x\).

Step 1: Factor the Numerators

First, we need to factor the numerators. Start with the numerator of the first fraction: \(x^2 - x - 6\).The factors of \(-6\) that add up to \(-1\) are \(2\) and \(-3\). Therefore, it factors into:\[(x - 3)(x + 2)\]Next, for the numerator of the second fraction: \(x^3 + x^2\).Factor out a common factor of \(x^2\):\[x^2(x + 1)\].

Step 2: Factor the Denominators

Now, factor the denominators. First, the denominator of the first fraction: \(x^2 + 2x\).Factor out a common factor of \(x\):\[x(x + 2)\].Next, the denominator of the second fraction: \(x^2 - 2x - 3\).The factors of \(-3\) that add up to \(-2\) are \(-3\) and \(1\). So it factors into:\[(x - 3)(x + 1)\].

Step 3: Rewrite the Expression

Now, rewrite the entire multiplication expression with the factored forms:\[\frac{(x - 3)(x + 2)}{x(x + 2)} \cdot \frac{x^2(x + 1)}{(x - 3)(x + 1)}\]

Step 4: Cancel Common Factors

Next, look for common factors that can be cancelled from the numerators and denominators:- The factor \(x + 2\) can be cancelled from the first fraction.- The factor \(x - 3\) can be cancelled from the entire expression.- The factor \(x + 1\) can be cancelled from the entire expression.After canceling, we are left with:\[\frac{x^2}{x}\]

Step 5: Simplify the Remaining Expression

Now, simplify the resulting expression. The remaining expression is \(\frac{x^2}{x}\).Cancel one \(x\) from both the numerator and denominator:\[x\].

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill to master when working with polynomial fractions. In this process, the goal is to express a polynomial as a product of its simplest expressions or factors. Take for example the polynomial \( x^2 - x - 6 \); we need to find two numbers that multiply to \(-6\) and add up to \(-1\). These numbers are \(2\) and \(-3\), allowing us to rewrite the polynomial as \((x - 3)(x + 2)\).

Similarly, when solving \( x^3 + x^2 \), look for the greatest common factor, which is \( x^2 \). By factoring \( x^2 \) out, we get \( x^2(x + 1) \). Factoring not only simplifies future calculations but also reveals opportunities to cancel out common polynomials in fractions.

Multiplication of Fractions

Multiplying fractions involves some straightforward but essential steps. First, multiply across the numerators; then, multiply across the denominators. In our exercise, the expression starts as \( \frac{(x - 3)(x + 2)}{x(x + 2)} \times \frac{x^2(x + 1)}{(x - 3)(x + 1)} \). Each of these fractions is already factored, making the next step easier.

Write the product of the numerators over the product of the denominators. Pay close attention to any factors that appear in both. These common factors will be key when simplifying the fractions later on. Multiplying fractions in a factored form eases the cancellation of terms and reduces the expression efficiently.

Simplifying Fractions

Simplifying fractions, especially polynomial ones, involves canceling common factors from numerators and denominators. After factoring, the expression is \( \frac{(x - 3)(x + 2)}{x(x + 2)} \cdot \frac{x^2(x + 1)}{(x - 3)(x + 1)} \). Start by identifying and canceling shared factors:

• \( x + 2 \)
• \( x - 3 \)
• \( x + 1 \)

By canceling these terms, you're left with \( \frac{x^2}{x} \). The last step is to simplify further by dividing both the numerator and denominator by \( x \), leading to \( x \).

This simplification reveals the true essence of the expression and makes it easier to understand or use in further solutions. Remember, simplified forms help reduce errors and clarify the mathematical relationships at play.