Polynomial Function: Zeros, Degree & Leading Coefficient

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Hey math enthusiasts! Let's dive into the fascinating world of polynomial functions. Today, we're going to crack the code on how to build a polynomial function when you're given some key ingredients: its degree, its zeros (also known as roots), and the leading coefficient. Specifically, we'll construct a degree 4 polynomial with specific zeros and a leading coefficient of 1. Ready to get started? Let's go!

Decoding the Problem: Our Blueprint

Okay, so what exactly are we dealing with? The problem throws a bunch of info at us, but don't worry, it's not as scary as it seems! We're tasked with finding a polynomial function, denoted as f(x). Here's what we know:

  • Degree: The degree of a polynomial tells us the highest power of the variable x. In our case, the degree is 4. This means our polynomial will have a term with x raised to the power of 4, and no higher powers.
  • Zeros (Roots): Zeros are the x-values where the function f(x) equals zero. They're the points where the graph of the polynomial crosses the x-axis. We have three zeros, but with different multiplicities. Multiplicity refers to how many times a particular zero appears as a root. Here's the breakdown:
    • x = -3, with a multiplicity of 2: This means the factor (x + 3) appears twice in the factored form of the polynomial.
    • x = 1, with a multiplicity of 1: This means the factor (x - 1) appears once.
    • x = 3, with a multiplicity of 1: This means the factor (x - 3) appears once.
  • Leading Coefficient: This is the number that multiplies the term with the highest power of x (the term with x to the power of 4 in our case). We're told the leading coefficient is 1. This simplifies things because we don't have to worry about scaling the entire function.

So, in essence, we have the building blocks: the degree dictates the overall shape, the zeros pinpoint where the graph kisses the x-axis, and the leading coefficient controls the vertical stretch or compression of the graph. Our mission? To assemble these pieces into a complete polynomial function!

To make sure we're on the right track, let's recap. We're looking for a polynomial function f(x) that:

  • Has a degree of 4
  • Has zeros at x = -3 (multiplicity 2), x = 1 (multiplicity 1), and x = 3 (multiplicity 1)
  • Has a leading coefficient of 1

Sounds good, right? Let's get to work! The process involves writing the polynomial in its factored form, which then allows us to expand and write it in standard form if we like.

Constructing the Factored Form: The Key to Unlocking the Polynomial

Alright, guys, let's get down to the nitty-gritty and build that factored form! The factored form of a polynomial is super useful because it directly shows us the zeros of the function. It’s like a secret decoder ring that reveals where the graph touches the x-axis. Remember those zeros and their multiplicities? They're our main tools here!

Here's how it works: for each zero, we create a factor. If a zero has a multiplicity greater than 1, we include that factor the corresponding number of times. Since our leading coefficient is 1, we don't need to add any additional constant multipliers at the beginning. This simplifies the process for us.

Let’s start with the first zero: x = -3. Since it has a multiplicity of 2, we’ll create the factor (x + 3) twice. This means we'll have (x + 3)(x + 3). Remember that when the zero is negative, the factor inside the parentheses will be positive, and vice versa. It’s a common point of confusion, so double-check! This ensures that when x = -3, each of the (x + 3) terms becomes zero, and the entire function becomes zero.

Next up, we have x = 1, with a multiplicity of 1. This gives us the factor (x - 1). The sign here matters, so watch out! When x = 1, the factor (x - 1) becomes zero.

Finally, we have x = 3, also with a multiplicity of 1. This provides the factor (x - 3). Again, the sign is important; when x = 3, the factor (x - 3) equals zero. We're on the right track!

Now, let's put it all together. The factored form of our polynomial function f(x) looks like this: f(x) = (x + 3)(x + 3)(x - 1)(x - 3). That's it! We have successfully created the factored form. This form is particularly helpful for sketching the graph of the polynomial. From the factored form, we can directly see the zeros of the function and their multiplicities. The leading coefficient also informs the end behavior of the polynomial's graph. We're so close to solving this! It’s like assembling the pieces of a puzzle. We have the factored form, and if we wanted, we could multiply everything out to get the standard form. But for now, we're good with the factored form.

Expanding for Standard Form (Optional): Putting It All Together

Okay, so we've got our factored form. But what if we want to see our polynomial function in a different light? What if we want to expand it and see it in standard form? Well, let's give it a try. This isn't strictly necessary, but it's a great exercise to solidify our understanding and get more comfortable working with polynomials. It’s like taking our creation, the factored form, and transforming it into something new!

Remember, the standard form of a polynomial is where the terms are arranged from the highest power of x down to the constant term. To get there, we’ll need to multiply out all the factors. This can seem a little tedious, but it's methodical, and we can do it! It's also a good way to double-check that we did the factored form right.

First, let's tackle the two (x + 3) factors. Multiplying these gives us (x + 3)(x + 3) = x^2 + 6x + 9. This is a common pattern, and you might even remember that (x + a)^2 = x^2 + 2ax + a^2.

Next, let’s bring in the (x - 1) and (x - 3) factors. We now have (x^2 + 6x + 9)(x - 1)(x - 3). We can start by multiplying (x^2 + 6x + 9) with (x - 1), which can be a bit intimidating. Let's do it carefully!

(x^2 + 6x + 9)(x - 1) = x^3 + 6x^2 + 9x - x^2 - 6x - 9 = x^3 + 5x^2 + 3x - 9

Finally, we need to multiply the result by (x - 3). So, we now have (x^3 + 5x^2 + 3x - 9)(x - 3). Multiplying this out carefully:

(x^3 + 5x^2 + 3x - 9)(x - 3) = x^4 + 5x^3 + 3x^2 - 9x - 3x^3 - 15x^2 - 9x + 27 = x^4 + 2x^3 - 12x^2 - 18x + 27

So, after all that multiplication, our polynomial in standard form is f(x) = x^4 + 2x^3 - 12x^2 - 18x + 27. Phew! We did it, guys! This is the same polynomial function as our factored form, just expressed differently.

While the factored form is handy for finding zeros, the standard form is useful for understanding the polynomial's overall shape, the y-intercept, and how the graph behaves for very large or very small x-values. Also, having the standard form lets us know for sure that we have a degree of 4, since the highest power of x is 4.

Conclusion: You've Got This!

And there you have it! We've successfully constructed a degree 4 polynomial function, given its zeros and leading coefficient. We started with the factored form, which clearly showed the zeros, and then optionally expanded it to standard form. This is a fundamental concept in algebra, so congratulations on leveling up your math skills!

Here’s a quick recap of the key takeaways:

  • The degree of a polynomial determines its overall shape.
  • The zeros (or roots) are the x-values where the function equals zero.
  • The leading coefficient affects the vertical stretch or compression.
  • The factored form makes it easy to spot the zeros, while the standard form is handy for general analysis.

Keep practicing, and you'll become a pro at these problems! Math can be a lot of fun, so don't be afraid to keep exploring and learning. Keep practicing the same way as we did today and you will be a math pro in no time! Remember, practice makes perfect!