Simplify Rational Expressions: Find Quotients & Exclusions

Hey there, math enthusiasts and problem-solvers! Ever stared at a complex rational division problem and wondered where to even begin? You're definitely not alone! Rational expressions, which are essentially fractions containing polynomials, might look intimidating at first glance, but I promise you, with the right approach and a few cool tricks up your sleeve, they become much more manageable – dare I say, fun? In this article, we're going to dive deep into a specific rational division problem: $\frac{-4 x-12}{x-7} \div \frac{x^2-12 x+36}{2 x-14}$. Our mission, should we choose to accept it, is twofold: first, we'll break it down piece by piece to find the quotient in lowest terms, and second, we'll figure out what values of x we absolutely, positively cannot allow in these expressions. Understanding these concepts isn't just about passing a math test; it's about building foundational skills that are super important in higher-level math, science, engineering, and even economics. So, grab your favorite beverage, get comfy, and let's unlock the secrets of rational division together. We're going to make sure every step is clear, every factor is accounted for, and every potential mathematical mishap is avoided. Ready? Let's roll!

Decoding Rational Division: A Step-by-Step Guide

Alright, guys, let's get down to business with our main challenge: $\frac{-4 x-12}{x-7} \div \frac{x^2-12 x+36}{2 x-14}$. When you encounter a rational division problem, your first and most crucial step is always factoring. Think of factoring as your algebraic superpower; it allows you to see the hidden components of each expression, which will be essential for simplification. Without proper factoring, you'll be trying to cancel terms that aren't truly factors, leading to incorrect results. It's like trying to untangle a knot without first understanding how the ropes are intertwined. So, let's break down each part of this problem, starting with our first fraction, then moving to the second, always keeping an eye out for common factors, special product patterns, or anything else that can make our lives easier.

Now, let's tackle the first rational expression: $\frac{-4 x-12}{x-7}$. The numerator, $-4x - 12$, immediately screams for factoring. Do you see a common factor there? Absolutely! Both terms are divisible by -4. So, we can pull out a -4, leaving us with $-4(x + 3)$. Simple enough, right? The denominator, $x-7$, is a prime polynomial in this context; it cannot be factored further into simpler linear terms with integer coefficients. This means it's already in its most basic form, which is totally fine! So, our first fraction, in its factored glory, becomes $\frac{-4(x+3)}{x-7}$. See? Not so scary when you break it down.

Next up, we have the second rational expression: $\frac{x^2-12 x+36}{2 x-14}$. Let's focus on the numerator first: $x^2 - 12x + 36$. Does this look familiar? It should! This is a classic example of a perfect square trinomial. Remember the pattern $(a - b)^2 = a^2 - 2ab + b^2$? Here, $a = x$ and $b = 6$. So, $x^2 - 12x + 36$ can be beautifully factored into $(x - 6)(x - 6)$, or simply $(x - 6)^2$. Pretty neat, huh? Now for its denominator, $2x - 14$. Again, we're looking for common factors. Both terms are clearly divisible by 2. Pulling out the 2, we get $2(x - 7)$. Fantastic! So, our second fraction transforms into $\frac{(x-6)^2}{2(x-7)}$.

With both fractions now neatly factored, our original division problem, $\frac{-4 x-12}{x-7} \div \frac{x^2-12 x+36}{2 x-14}$, becomes $\frac{-4(x+3)}{x-7} \div \frac{(x-6)^2}{2(x-7)}$. And here comes the magic step for any rational division: "Keep, Change, Flip!" Or, more formally, to divide by a fraction, you multiply by its reciprocal. This means we keep the first fraction as is, change the division sign to multiplication, and flip (find the reciprocal of) the second fraction. So, our expression transforms into $\frac{-4(x+3)}{x-7} \cdot \frac{2(x-7)}{(x-6)^2}$. This conversion from division to multiplication is absolutely fundamental and opens the door for easy simplification.

Now that it's a multiplication problem, the fun really begins! We can look for common factors in the numerators and denominators across both fractions. It's like a scavenger hunt! We have $\frac{-4(x+3)}{x-7} \cdot \frac{2(x-7)}{(x-6)^2}$. Notice anything that appears in both a numerator and a denominator? Bingo! The term $(x - 7)$ is in the denominator of the first fraction and the numerator of the second. This is a common factor, and we can cancel it out! When we cancel $(x - 7)$ from both positions, we're left with $\frac{-4(x+3)}{1} \cdot \frac{2}{(x-6)^2}$. Now, multiply the remaining numerators together and the remaining denominators together. The numerators become $-4(x+3) \cdot 2 = -8(x+3)$. The denominators become $1 \cdot (x-6)^2 = (x-6)^2$. Putting it all together, our simplified expression is $\frac{-8(x+3)}{(x-6)^2}$. We've meticulously factored, flipped, and canceled our way to a much simpler form. This entire process relies on careful algebraic manipulation, ensuring that no potential factors are overlooked, and every step logically follows the rules of rational expression arithmetic.

The Quotient in Lowest Terms: Unveiling the Simplified Expression

After all that meticulous factoring, flipping, and strategically canceling common factors, we've finally arrived at our simplified quotient! The quotient of the given rational division problem, in its absolute lowest terms, is $\frac{-8(x+3)}{(x-6)^2}$. Isn't that a thing of beauty? From the initial complex beast, we've distilled it down to this elegant and much more manageable expression. The reason we call this