Verified Answer :

  • Convert the division of the two rational expressions into a multiplication by the reciprocal of the divisor.
  • Factor the polynomials in the expressions to identify common factors: \(3x-6 = 3(x-2)\) and \(x^2-x-2 = (x-2)(x+1)\).
  • Cancel common factors such as \((x-2)\) from the numerator and denominator.
  • Combine the remaining terms to reach the final simplified form, \(\boxed{\frac{x^2+3x+2}{3x-9}}\).

Explanation

  1. Rewriting the Division as Multiplication

To simplify the given rational expression \(\frac{x+2}{3x-6} \div \frac{x-3}{x^2-x-2}\), we start by rewriting the division as multiplication by the reciprocal of the divisor. Division by a fraction is the same as multiplying by its reciprocal:
\[\frac{x+2}{3x-6} \cdot \frac{x^2-x-2}{x-3}\]
For this expression to be defined, the denominators must not be zero:

  • \(3x-6 eq 0 \implies x eq 2\)
  • \(x-3 eq 0 \implies x eq 3\)
  • \(x^2-x-2 eq 0 \implies (x-2)(x+1) eq 0 \implies x eq 2, x eq -1\)

Combining these, we must have \(x eq 2, 3, -1\).

  1. Factoring the Rational Expressions

Next, we factor the polynomials in the numerators and denominators to identify any common factors.

  • The denominator \(3x-6\) can be factored as \(3(x-2)\).
  • The quadratic numerator \(x^2-x-2\) can be factored by looking for two numbers that multiply to \(-2\) and add to \(-1\). These are \(-2\) and \(1\), so \(x^2-x-2 = (x-2)(x+1)\).

Now, substitute these factored forms back into our expression:
\[\frac{x+2}{3(x-2)} \cdot \frac{(x-2)(x+1)}{x-3}\]

  1. Simplifying and Finalizing the Expression

Now we can simplify the expression by canceling common factors found in both the numerator and the denominator. We see that the term \((x-2)\) appears in the numerator and the denominator:
\[\frac{x+2}{3(x-2)} \cdot \frac{(x-2)(x+1)}{x-3} = \frac{(x+2) \cdot (x+1)}{3 \cdot (x-3)}\]
We can multiply the terms to obtain the final simplified form:
\[\frac{(x+2)(x+1)}{3(x-3)} = \frac{x^2+x+2x+2}{3x-9} = \frac{x^2+3x+2}{3x-9}\]
Thus, the simplified expression is \(\boxed{\frac{x^2+3x+2}{3x-9}}\).

Examples

Rational expressions are commonly used in engineering and physics when calculating rates, such as average speed or fluid flow through pipes. For instance, if you are designing a system where the total output of a process (represented by a polynomial) needs to be distributed across several components (represented by another polynomial), simplifying these expressions allows you to predict system behavior more efficiently. By reducing complex formulas to their simplest form, engineers can quickly assess how changes in one variable, like the diameter of a pipe or the flow pressure, affect the entire system's performance.

Related Questions