Finding The Polynomial Function: A Step-by-Step Guide

Hey guys, let's dive into the fascinating world of polynomial functions! Today, we're tackling a cool problem in mathematics where we're given some clues about a polynomial function, and our mission is to figure out a possible expression for it. Specifically, we're dealing with a function named m, which is a polynomial function of degree 4. This means the highest power of x in our function is 4. We're also given some super important information: the zeros of m. Remember, zeros are the x-values where the function equals zero. In our case, the zeros of m include 0, 5, and 3 - 2i. This is going to be fun, and a bit challenging too, as it requires us to remember some important concepts of polynomial and complex numbers.

Now, let's break down how we can crack this problem. We'll explore the properties of polynomial functions, how to use the given zeros to build the function, and some neat tricks along the way. Get ready to flex those math muscles and see how all the pieces fit together. This journey will be full of mathematics, which is full of interesting problems like this.

Unveiling the Secrets of Polynomial Functions

Alright, before we get our hands dirty with the specific function m, let's take a quick pit stop to recap some key ideas about polynomial functions. First off, what exactly is a polynomial function? Well, it's a function that can be written in the form:

p(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0*

where a_n, a_{n-1}, ..., a_1, a_0 are coefficients (real numbers) and n is a non-negative integer. The degree of the polynomial is the highest power of x with a non-zero coefficient (in our case, it's 4). The zeros of a polynomial function are the values of x that make the function equal to zero. These zeros are super crucial because they tell us a lot about the function's behavior and structure. When you see a zero, say r, it means that (x - r) is a factor of the polynomial. So, if we know the zeros, we can start building the polynomial using these factors.

For our function m, we know it has a degree of 4. This means our final expression for m(x) will have an x term raised to the fourth power. Also, we know that the zeros include 0, 5, and 3 - 2i. Since the degree is 4, we are missing one more zero. But don't worry, we'll get to that. The given zeros give us a starting point to construct our polynomial.

One more important thing, especially for polynomials with real coefficients: complex zeros always come in conjugate pairs. This means if 3 - 2i is a zero, then its conjugate, 3 + 2i, must also be a zero. With this in mind, we have all the pieces we need to find a possible expression for m(x). We can write the polynomial as a product of linear factors, using the zeros we've identified. And with the help of the conjugate pair rule, we'll see that it's not so hard after all!

Constructing the Expression for m(x)

Okay, time to put our knowledge to work and actually build the expression for m(x). We've got our zeros: 0, 5, 3 - 2i, and 3 + 2i. As we discussed earlier, each zero corresponds to a linear factor of the form (x - r). So, let's turn our zeros into factors:

• For the zero 0, the factor is (x - 0) = x
• For the zero 5, the factor is (x - 5)
• For the zero 3 - 2i, the factor is (x - (3 - 2i)) = (x - 3 + 2i)
• For the zero 3 + 2i, the factor is (x - (3 + 2i)) = (x - 3 - 2i)

Now, we multiply these factors together. Remember that since the degree of m(x) is 4, we need to have four factors. So, our expression for m(x) will be:

m(x) = ax*(x - 5)(x - 3 + 2i)(x - 3 - 2i)*

Here, a is a constant that can take any non-zero real value. It's there because multiplying the whole polynomial by a constant doesn't change the zeros, but it does affect the leading coefficient and the overall shape of the graph. Often, we simply take a = 1 to find a possible expression. Let's simplify and make things look nicer. First, we'll deal with those complex conjugate factors:

(x - 3 + 2i)(x - 3 - 2i)* = ((x - 3) + 2i)((x - 3) - 2i)*

This is a classic form that can be simplified using the difference of squares formula, (a + b)(a - b) = a^2 - b^2. So, we get:

(x - 3)^2 - (2i)^2 x^2 - 6x + 9 - (-4) x^2 - 6x + 13

Now, we can substitute this back into our expression for m(x):

m(x) = ax*(x - 5)(x^2 - 6x + 13)

If we let a = 1, we have a possible expression for m(x). We can leave it like this, or we can expand and combine like terms. If we expand, we will get:

m(x) = x(x - 5)(x^2 - 6x + 13) m(x) = (x^2 - 5x)(x^2 - 6x + 13)* m(x) = x^4 - 6x^3 + 13x^2 - 5x^3 + 30x^2 - 65x m(x) = x^4 - 11x^3 + 43x^2 - 65x

Either the factored form or the expanded form is a correct answer. However, the factored form is easier to read and allows you to quickly see the zeros of the polynomial. Therefore, the expression can be any of the two forms. Let's make sure you get the gist of it.

Unpacking Complex Conjugate Zeros

Let's pause for a moment to really understand why complex zeros always come in conjugate pairs, like 3 - 2i and 3 + 2i. This is super important because it ensures that our polynomial function has real coefficients. If a polynomial has real coefficients, then the imaginary parts of the complex zeros must cancel each other out when we multiply out the factors. Let's see how this works using our example. The factors corresponding to our complex zeros are (x - (3 - 2i)) and (x - (3 + 2i)). When we multiply these factors, we get:

(x - 3 + 2i)(x - 3 - 2i) = (x - 3)^2 - (2i)^2 = x^2 - 6x + 9 - (-4) = x^2 - 6x + 13*

Notice how the imaginary terms disappear! This is because the i terms cancel each other out, leaving us with a quadratic expression that has only real coefficients. If we didn't have the conjugate pair, we would end up with imaginary coefficients, which is not what we want if we're dealing with a polynomial function with real coefficients. This is the power of the conjugate pair theorem in action! This is why, when we find a complex zero, we automatically know its conjugate is also a zero. It's a handy trick that helps us build the correct polynomial expression.

Putting it All Together: The Final Expression

Alright, we've done all the hard work, guys! Let's summarize and write down our final expression for m(x). We started with the zeros: 0, 5, 3 - 2i, and 3 + 2i. We used these zeros to create the factors and then multiplied them together. After simplifying, we got two possible expressions, one factored and one expanded. Remembering that the leading coefficient can be any real non-zero number, we'll set a = 1 for simplicity. So, here's our final answer:

• Factored form: m(x) = x(x - 5)*(x^2 - 6x + 13)*
• Expanded form: m(x) = x^4 - 11x^3 + 43x^2 - 65x

Either expression is a possible answer. The first one is the best because it clearly shows the zeros of the function, while the second one is the standard form of the polynomial function. We have successfully found a possible expression for the polynomial function m given its degree and zeros! Awesome work, everyone! You've learned how to use the zeros, including complex conjugate pairs, to build a polynomial expression. We've seen how the degree of the polynomial affects the number of factors and how the complex conjugate theorem ensures real coefficients. This knowledge is going to come in handy in the future, as you continue to explore the world of mathematics. Keep up the great work and keep exploring! Congratulations, we're done.