91Ó°ÊÓ

Factoring is an essential technique in simplifying rational expressions. To factor a polynomial, you need to find expressions that multiply to give the original polynomial. This often involves finding two binomials whose product equals the polynomial. For example, consider the polynomial \(p^2 - 11p + 30\). By examining the polynomial, we look for two numbers that multiply to 30 and add up to -11. These numbers are -5 and -6, so we can factor it as \((p-5)(p-6)\).
Similarly, for \(p^2 - 2p - 24\), we seek two numbers that multiply to -24 and add to -2. Here, we find -6 and 4, so the factorization is \((p-6)(p+4)\).
Polynomials like \(p^2 + 5p + 4\) can be factored into \((p+1)(p+4)\) by looking for numbers that multiply to 4 and add up to 5. Understanding how to factor polynomials helps break down complex expressions into simpler, more manageable parts.

After factoring polynomials, the next step in simplifying rational expressions is to cancel common factors. Common factors in the numerator and denominator can be canceled out because any number divided by itself is 1. For instance, in the fraction \((p-5)(p-6) / (p-6)(p+4)\), the \(p-6\) terms in the numerator and denominator can be canceled. This simplification makes the fraction easier to handle.
In our original exercise, after factoring, we have \((p-5)(p-6) / (p-6)(p+4)\) and \((p-5)(p+1) / (p+1)(p+4)\). Here, \(p-6\) in the first fraction and \(p+1\) in the second can be canceled. This step is crucial in reducing complexity and making further calculations more straightforward.

Multiplying rational expressions involves flipping the second fraction (taking its reciprocal) and then multiplying. For our exercise, we transform the division of fractions into multiplication by flipping the second fraction. We start with: \left( \frac{(p-5)/(p+4)}{(p-5)/(p+4)} \right)\.
We convert this into multiplication: \left( \frac{(p-5)}{(p+4)} \times \frac{(p+4)}{(p-5)} \right)\.
During multiplication, we multiply numerators together and denominators together. In our exercise, after converting the division into multiplication, we have: \left( \frac{(p-5)}{(p+4)} \times \frac{(p+4)}{(p-5)} \right)\. Canceling the common factors in the resulting expression \(p-5\) and \(p+4\), we ultimately get: 1.
By understanding these steps - factoring, canceling, and multiplying, students can simplify complex rational expressions with confidence.