Polynomial Mastery: Finding A Degree 4 Function

Hey math enthusiasts! Let's dive into the fascinating world of polynomials. Today's mission: to find a degree 4 polynomial, $f(x)$, with real coefficients, given some crucial clues about its zeros. We've got a couple of repeat offenders and a complex number in the mix. Sounds like a fun challenge, right? The given zeros are: -1 (with a multiplicity of 2) and 1 - 3i. So, buckle up, grab your pencils, and let's get started. We'll break down the steps, explain the reasoning, and hopefully make the whole process crystal clear. This is a great opportunity to flex our algebra muscles and truly understand how polynomials behave. Understanding polynomial functions is absolutely fundamental in mathematics, extending far beyond the classroom into fields like engineering, computer science, and physics. Being able to construct and analyze these functions is a key skill. Let's start with a refresher on what we know about polynomials. A polynomial of degree n will have n roots (or zeros), though some can repeat. The Fundamental Theorem of Algebra is the king of this castle, basically guaranteeing that our degree 4 polynomial will have exactly 4 roots (counting multiplicities). Also, the real coefficients part is super important, because it implies something special about our complex roots.

Remember, a root (or a zero) is a value of x that makes the function equal to zero, i.e., $f(x) = 0$. The multiplicity of a root tells us how many times that particular root appears. So, in our case, -1 appears twice. Also, we are dealing with a degree 4 polynomial; that is, the highest power of x will be 4.

We will go step by step with the process and at the end of it, we will have our answer. Ready?

Unveiling the Zeros: The Building Blocks

Alright, let's list down what we already know. We have the following zeros:

• -1 (with multiplicity 2): This means that (x + 1) is a factor twice. So, we'll have $(x + 1)^2$ in our polynomial.
• 1 - 3i: This is a complex root. Now, here's a crucial rule: Complex roots always come in conjugate pairs when the polynomial has real coefficients. This is a big deal! It means if 1 - 3i is a root, then its conjugate, 1 + 3i, is also a root. The complex conjugate of a number is formed by changing the sign of its imaginary part. So, if we have $a + bi$, its conjugate is $a - bi$. This conjugate-pair rule is really important; we'll use it to our advantage in a bit. So, now we have all four roots: -1, -1, 1 - 3i, and 1 + 3i. Awesome! That's all we need to start constructing our function.

This principle helps ensure that our polynomial has real coefficients, as any imaginary parts introduced by complex roots will cancel out when we expand the polynomial. By identifying the conjugate pair, we now have a complete set of roots that aligns with the degree of our polynomial. These roots serve as the foundation upon which we will build our equation, allowing us to accurately represent the polynomial's behavior and characteristics. The next step is to use these zeros to get the factors of our polynomial. Let's do it!

Constructing the Factors: From Zeros to Expressions

Now that we have all four zeros, we can write the factors of our polynomial. Remember, if 'r' is a zero, then (x - r) is a factor.

• For -1 (multiplicity 2), we get the factors: $(x + 1)$ and $(x + 1)$ again. So, this gives us $(x + 1)^2$ or $(x^2 + 2x + 1)$.
• For 1 - 3i, we get the factor: $(x - (1 - 3i)) = (x - 1 + 3i)$.
• For 1 + 3i, we get the factor: $(x - (1 + 3i)) = (x - 1 - 3i)$.

So, our polynomial will have the form: $f(x) = a(x + 1)^2(x - 1 + 3i)(x - 1 - 3i)$, where 'a' is a real constant. Since we're not explicitly given any other point on the curve, we will assume $a = 1$.

Now, let's simplify our factors, and put them together. The goal here is to get an expression for our function. You'll see that, because of the complex conjugates, the imaginary parts will magically disappear when we multiply everything out. This will leave us with real coefficients, as it should. We'll start by multiplying the complex conjugate factors together.

It's time to put on our multiplication hats, and start constructing our function. This is where we take the factors and turn them into a fully-fledged polynomial. The trickiest part will be multiplying the complex conjugate factors. This will require some careful distribution, but we can do it!

Multiplying It Out: The Grand Finale

Let's get down to the nitty-gritty and multiply the factors together to find our polynomial $f(x)$. First, let's expand the $(x + 1)^2$ factor, which we've already done, so we know it's $(x^2 + 2x + 1)$. Next, we will multiply the complex conjugate factors: $(x - 1 + 3i)(x - 1 - 3i)$. This might look a little daunting, but we'll take it step by step.

$(x - 1 + 3i)(x - 1 - 3i) = x(x - 1 - 3i) - 1(x - 1 - 3i) + 3i(x - 1 - 3i)$.

$= x^2 - x - 3ix - x + 1 + 3i + 3ix - 3i - 9i^2$.

Notice how the $-3ix$ and $+3ix$ terms cancel out. Also, we know that $i^2 = -1$. So, the expression simplifies to:

$= x^2 - 2x + 1 - 9(-1)$.

$= x^2 - 2x + 1 + 9$.

$= x^2 - 2x + 10$.

Now we have:

$f(x) = (x^2 + 2x + 1)(x^2 - 2x + 10)$.

Now, let's multiply those two quadratics together:

$f(x) = x^2(x^2 - 2x + 10) + 2x(x^2 - 2x + 10) + 1(x^2 - 2x + 10)$.

$= x^4 - 2x^3 + 10x^2 + 2x^3 - 4x^2 + 20x + x^2 - 2x + 10$.

Combine like terms:

$f(x) = x^4 + (-2x^3 + 2x^3) + (10x^2 - 4x^2 + x^2) + (20x - 2x) + 10$.

$f(x) = x^4 + 7x^2 + 18x + 10$.

And there we have it! The final answer! We've successfully constructed a degree 4 polynomial, with real coefficients, that satisfies all the conditions of our problem. We found that $f(x) = x^4 + 7x^2 + 18x + 10$. We've shown how the complex conjugate pairs work together to ensure that our polynomial has real coefficients. The entire process is really just a systematic application of the relationships between the zeros of a polynomial and its factored form.

Conclusion: Polynomial Power Unleashed

Fantastic job, everyone! We've successfully navigated the process of finding a degree 4 polynomial with real coefficients, given its zeros, which included a complex conjugate pair. We saw how important the conjugate pair concept is, and how it helps us construct polynomials with real coefficients. This journey underscores a fundamental concept in algebra: how the roots of a polynomial determine its behavior and form. Remember, the process involves understanding zeros, constructing factors, and skillfully expanding the factors to get the polynomial. We used the concepts of multiplicity and conjugate pairs. We carefully multiplied the factors, paying attention to the details, and simplifying step-by-step. In the end, we achieved our goal. This is a common type of problem in algebra and is a fundamental skill in mathematics. Now, armed with this knowledge, you can tackle similar problems with confidence. Keep practicing, keep exploring, and keep the math adventures going! Well done, guys!