Roots Of F(x) = (x-6)^2(x+2)^2: A Step-by-Step Solution

by Omar Yusuf 56 views

Hey guys! Today, we're diving into a fun math problem: finding the roots of the polynomial function f(x) = (x-6)2(x+2)2. It might seem a little daunting at first, but don't worry, we'll break it down step by step. We'll not only find the roots but also explore the concept of multiplicity, which is super important for understanding the behavior of polynomial functions. So, buckle up and let's get started!

Understanding Roots and Multiplicity

Before we jump into solving our specific problem, let's make sure we're all on the same page about what roots and multiplicity actually mean. This foundational knowledge is crucial for tackling any polynomial equation.

What are Roots?

The roots of a function, also known as zeros or solutions, are the values of x that make the function equal to zero. In other words, they are the points where the graph of the function intersects the x-axis. Finding the roots is a fundamental problem in algebra and calculus, and it has applications in various fields, from physics and engineering to economics and computer science. Think of it like finding the key that unlocks the equation and makes it true. When f(x) equals zero, we've found one of those keys.

What is Multiplicity?

Now, here's where it gets a little more interesting. The multiplicity of a root refers to the number of times that root appears as a factor in the polynomial. Let's say we have a factor like (x - a)^n. Here, 'a' is the root, and 'n' is its multiplicity. The multiplicity tells us something important about how the graph of the function behaves at that root. If the multiplicity is 1, the graph crosses the x-axis at that root. If the multiplicity is an even number (like 2, 4, etc.), the graph touches the x-axis but doesn't cross it – it "bounces" off the x-axis. If the multiplicity is an odd number greater than 1 (like 3, 5, etc.), the graph flattens out as it crosses the x-axis. Understanding multiplicity helps us visualize the graph of a polynomial function more accurately.

In simpler terms, multiplicity is like the strength of the root. A higher multiplicity means the root has a greater influence on the function's behavior near that point. It's not just about where the graph hits the x-axis, but how it hits the x-axis. This is a critical concept for sketching polynomial graphs and understanding their properties. Remember this distinction: crossing the x-axis (multiplicity 1), bouncing off (even multiplicity), and flattening out (odd multiplicity greater than 1).

Solving for the Roots of f(x) = (x-6)2(x+2)2

Okay, now that we've got the definitions down, let's get our hands dirty with the actual problem! We're going to find the roots of the function f(x) = (x-6)2(x+2)2. Remember, the goal is to find the values of x that make f(x) equal to zero. This means we need to set the entire expression equal to zero and solve for x.

The beauty of this particular function is that it's already factored for us! This makes our job significantly easier. We have a product of two squared terms: (x-6)^2 and (x+2)^2. For the entire product to be zero, at least one of these terms must be zero. This is a fundamental principle in algebra: if A * B = 0, then either A = 0 or B = 0 (or both).

So, let's set each factor equal to zero and solve:

  1. (x - 6)^2 = 0

To solve this, we can take the square root of both sides: √(x - 6)^2 = √0. This gives us x - 6 = 0. Adding 6 to both sides, we find our first root: x = 6.

  1. (x + 2)^2 = 0

Similarly, we take the square root of both sides: √(x + 2)^2 = √0. This gives us x + 2 = 0. Subtracting 2 from both sides, we find our second root: x = -2.

So, we've found two roots: x = 6 and x = -2. But we're not done yet! We need to determine the multiplicity of each root. Remember, the multiplicity is the exponent of the factor that produces the root. In this case, both factors are squared, meaning the exponent is 2 for each.

Therefore, the root x = 6 has a multiplicity of 2, and the root x = -2 also has a multiplicity of 2. This is a key finding! It tells us that the graph of this function will "bounce" off the x-axis at both x = 6 and x = -2, rather than crossing it. Understanding the multiplicity helps us visualize the shape of the graph without even needing to plot points.

Determining the Multiplicity

Now that we've identified the roots (x = 6 and x = -2), the next crucial step is to pinpoint their multiplicities. Multiplicity, as we discussed earlier, plays a pivotal role in understanding how the graph of the function behaves around these roots. Think of it as the secret ingredient that reveals the graph's personality at these critical points.

Looking back at our function, f(x) = (x-6)2(x+2)2, the multiplicities are staring right at us! They're the exponents on the factors. Let's break it down:

  • The factor (x - 6)^2: The exponent here is 2. This tells us that the root x = 6 has a multiplicity of 2. A multiplicity of 2 means the graph will touch the x-axis at x = 6 but not cross it. It's like a gentle kiss – the graph makes contact but then turns away.

  • The factor (x + 2)^2: Similarly, the exponent here is also 2. This means the root x = -2 has a multiplicity of 2 as well. Just like at x = 6, the graph will touch the x-axis at x = -2 and bounce back. This consistent behavior due to the even multiplicity creates a symmetrical feel around these roots.

So, we've confidently determined that both roots, x = 6 and x = -2, have a multiplicity of 2. This knowledge is powerful. We can now visualize the graph as a curve that approaches the x-axis at x = 6 and x = -2, gently touches it, and then turns back in the same direction. This bouncing behavior is a direct consequence of the even multiplicity.

Choosing the Correct Answer

Alright, we've done the hard work! We've found the roots and their multiplicities. Now it's time to choose the correct answer from the given options. Remember, our roots are:

  • x = 6 with multiplicity 2
  • x = -2 with multiplicity 2

Let's go through the options:

A. -6 with multiplicity 1: Incorrect. We found a root of 6, not -6, and its multiplicity is 2, not 1.

B. -6 with multiplicity 2: Incorrect. Again, the root is 6, not -6.

C. 6 with multiplicity 1: Incorrect. The root is correct, but the multiplicity is 2, not 1.

D. 6 with multiplicity 2: Correct! This matches our finding.

E. -2 with multiplicity 1: Incorrect. The root is correct, but the multiplicity is 2, not 1.

F. -2 with multiplicity 2: Correct! This also matches our finding.

G. 2 with multiplicity 1: Incorrect. We found a root of -2, not 2.

H. 2 with multiplicity 2: Incorrect. The root is -2, not 2.

Therefore, the correct answers are D and F. We've successfully identified the roots and their multiplicities, and we've matched them to the correct options. Pat yourself on the back – you've conquered this polynomial problem!

Visualizing the Graph (Optional)

Want to take your understanding a step further? Let's visualize the graph of f(x) = (x-6)2(x+2)2. We already know some key information:

  • Roots: x = 6 and x = -2
  • Multiplicities: Both roots have a multiplicity of 2, meaning the graph bounces off the x-axis at these points.

To get a general idea of the shape, we can also consider the leading term of the polynomial. If we were to expand the function, the leading term would be x^4 (because we have x^2 * x^2). This means the graph will have a general "U" shape, opening upwards. The ends of the graph will point upwards as x approaches positive or negative infinity.

Knowing this, we can sketch a rough graph. It will be a curve that touches the x-axis at x = -2 and x = 6, bouncing off at both points, and opening upwards on both ends. There will be a minimum point somewhere between x = -2 and x = 6. While we haven't calculated the exact minimum point, we have a pretty good idea of the overall shape thanks to our understanding of roots and multiplicity. You can use graphing software or online tools to plot the function and see the exact shape. It's a great way to reinforce your understanding!

Conclusion

So, there you have it! We've successfully found the roots of f(x) = (x-6)2(x+2)2, determined their multiplicities, and even visualized the graph. Remember, the key takeaways are:

  • Roots are the values of x that make the function equal to zero.
  • Multiplicity tells us how many times a root appears as a factor and how the graph behaves at that root.
  • Even multiplicities mean the graph bounces off the x-axis.

Understanding these concepts will help you tackle a wide range of polynomial problems. Keep practicing, and you'll become a root-finding pro in no time! Keep exploring and happy problem-solving!