Finding Roots At Which The Graph Touches The X-Axis For F(x)=(x-5)³(x+2)²

Hey there, math enthusiasts! Today, we're diving into a fascinating problem involving polynomial functions and their graphs. Specifically, we're going to figure out at which root the graph of the function f(x) = (x - 5)³(x + 2)² touches the x-axis. This is a classic concept in algebra, and understanding it will give you a solid grasp of how polynomial functions behave. Let's break it down step by step, making sure everyone's on board.

Understanding Roots and Their Significance

To really nail this, we first need to understand what roots are and what it means for a graph to “touch” the x-axis. In math lingo, roots (also called zeros or x-intercepts) are the values of x that make the function f(x) equal to zero. Graphically, these are the points where the curve intersects or touches the x-axis. Finding the roots is crucial for sketching the graph of a polynomial because they serve as key anchor points. Think of roots as the foundation upon which the entire graph is built. For our function, f(x) = (x - 5)³(x + 2)², setting each factor to zero gives us the roots. So, we have (x - 5)³ = 0 and (x + 2)² = 0. Solving these simple equations, we find that x = 5 and x = -2 are our roots. But hold on, it's not just about finding the roots; it's also about understanding their multiplicity, which is where the touching versus crossing part comes in.

Multiplicity: The Key to Touching vs. Crossing

Now comes the really cool part – understanding multiplicity. The multiplicity of a root is the number of times the corresponding factor appears in the factored form of the polynomial. It tells us a lot about how the graph behaves at that root. In our case, the factor (x - 5) appears three times (because of the exponent 3), so the root x = 5 has a multiplicity of 3. The factor (x + 2) appears twice (exponent 2), so the root x = -2 has a multiplicity of 2. This is where the magic happens. If a root has an odd multiplicity (like 1, 3, 5, etc.), the graph crosses the x-axis at that root. If a root has an even multiplicity (like 2, 4, 6, etc.), the graph touches the x-axis and turns around. It's like the graph kisses the x-axis and then bounces back. So, for our function, the graph will cross the x-axis at x = 5 (multiplicity 3) and touch the x-axis at x = -2 (multiplicity 2). The multiplicity acts like a secret code, revealing the graph's behavior at each root. Isn't that neat?

Visualizing the Graph's Behavior

Let's visualize this to make it crystal clear. Imagine the x-axis as a road and our graph as a car driving along it. When the car approaches x = 5, it's like it hits a bump in the road, crosses over to the other side, and keeps going. This is the crossing behavior due to the odd multiplicity. But when the car approaches x = -2, it's like it hits a wall, gently touches it, and then turns around and heads back in the direction it came from. This is the touching behavior due to the even multiplicity. Think of it this way: odd multiplicity means a full crossing, while even multiplicity means a gentle touch and turn. This visualization helps solidify the concept and makes it easier to remember. You can even sketch a rough graph to see this in action – plot the roots on the x-axis, and then imagine the curve crossing at x = 5 and touching and turning at x = -2. The overall shape of the graph will also depend on the leading coefficient and the degree of the polynomial, but the behavior at the roots is primarily determined by their multiplicities.

Analyzing the Given Function: f(x) = (x - 5)³(x + 2)²

Now, let’s get back to our specific problem. We're dealing with the function f(x) = (x - 5)³(x + 2)². As we've already established, the roots are x = 5 and x = -2. The root x = 5 comes from the factor (x - 5)³, which has a multiplicity of 3 (odd). This means the graph crosses the x-axis at x = 5. The root x = -2 comes from the factor (x + 2)², which has a multiplicity of 2 (even). This means the graph touches the x-axis at x = -2. Therefore, the answer to our question—at which root does the graph touch the x-axis—is x = -2. By carefully examining the factors and their exponents, we can directly determine where the graph touches versus crosses the x-axis. This technique is super useful for quickly analyzing polynomial functions without needing to graph them perfectly. You can impress your friends with this math magic!

Putting It All Together

So, to recap, finding the roots of a polynomial is just the first step. Understanding the multiplicity of each root is what gives us deeper insight into the graph's behavior. Even multiplicities mean the graph touches the x-axis, while odd multiplicities mean it crosses. For the function f(x) = (x - 5)³(x + 2)², the graph touches the x-axis at the root x = -2 because it has an even multiplicity of 2. This is a fundamental concept in algebra and is essential for anyone wanting to master polynomial functions. Think of it as decoding the secret language of graphs! By grasping these concepts, you'll be able to quickly analyze and sketch polynomial functions, making you a true math whiz.

Identifying the Correct Option

Given our analysis, we can now confidently identify the correct answer from the options provided: A. -5, B. -2, C. 2, and D. 5. We've determined that the graph of f(x) = (x - 5)³(x + 2)² touches the x-axis at x = -2. Therefore, the correct answer is B. -2. It's always satisfying when the hard work pays off and we can pinpoint the right answer. Remember, it's not just about getting the answer; it's about understanding the why behind it. That's what truly cements the knowledge and makes you a better problem-solver.

Final Thoughts and Tips

In conclusion, understanding roots and their multiplicities is crucial for analyzing polynomial functions. The multiplicity tells us whether the graph crosses or touches the x-axis at a particular root. Remember, even multiplicities mean touching, and odd multiplicities mean crossing. This knowledge is invaluable for sketching graphs and solving problems related to polynomial behavior. Keep practicing these concepts, and you'll become a pro at analyzing graphs in no time! Math can be challenging, but it's also incredibly rewarding when you unlock its secrets. So, keep exploring, keep questioning, and most importantly, keep having fun with it!

Additional Tips for Success

To really master this topic, here are a few extra tips: Practice graphing polynomial functions with different multiplicities. Try varying the exponents and observe how the graph changes at each root. Use graphing calculators or online tools to visualize these functions. Experiment with different polynomials to see the effects of different roots and multiplicities. This hands-on experience will greatly enhance your understanding. Also, don’t hesitate to review the basics of factoring polynomials. Being able to quickly factor a polynomial is key to finding its roots. The more you practice, the more these concepts will become second nature. Finally, remember to stay curious and ask questions. Math is a journey, and every question is a step forward. By consistently applying these tips and maintaining a curious attitude, you'll be well on your way to mastering polynomial functions and beyond. So go ahead, tackle those problems with confidence, and remember – you've got this!