Constructing A Polynomial Function Fifth Degree With Given Zeros And Leading Coefficient

Hey guys! Let's dive into the fascinating world of polynomials and figure out how to build one from scratch! In this article, we're going to tackle a classic problem: constructing a polynomial function with some very specific characteristics. We're talking about a fifth-degree polynomial that has a zero at 2 with a multiplicity of 4, another zero at 4, and a leading coefficient of 2. Sounds like a fun puzzle, right? Let’s break it down step by step.

Understanding the Problem

Before we jump into the nitty-gritty, let’s make sure we’re all on the same page. The problem asks us to create a polynomial function, which is basically an expression with variables and coefficients like $f(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f$. We've got some clues to help us out:

• Fifth Degree: This tells us the highest power of $x$ in our polynomial will be 5. So, our function will look something like $f(x) = ax^5 + bx^4 + ...$.
• 2 is a Zero of Multiplicity 4: This is a big one! A zero is a value of $x$ that makes the polynomial equal to zero. When we say 2 is a zero with multiplicity 4, it means $(x - 2)$ appears as a factor four times. Think of it as $(x - 2)^4$ being part of our polynomial.
• 4 is the Only Other Zero: This means that $x = 4$ is another value that makes our polynomial zero. So, $(x - 4)$ will also be a factor.
• Leading Coefficient is 2: The leading coefficient is the number in front of the highest power of $x$. In our case, it's the 'a' in $ax^5$, and we know it's equal to 2.

Now that we've decoded the clues, let's put them together to construct our polynomial.

Building the Polynomial

Okay, so how do we turn these clues into an actual function? It's like piecing together a puzzle. We'll start with the zeros and their multiplicities.

Incorporating the Zeros

We know that 2 is a zero of multiplicity 4. This means the factor $(x - 2)$ appears four times in our polynomial. So we have $(x - 2)^4$. This part is crucial because it ensures that when $x = 2$, the entire term becomes zero, satisfying our condition.

We also know that 4 is another zero. This gives us a factor of $(x - 4)$. Simple enough, right? When $x = 4$, this term becomes zero, just as we need.

Now, let's combine these factors. We have:

$(x - 2)^4(x - 4)$

This expression already gives us a polynomial with the correct zeros and multiplicities. But remember, we need a fifth-degree polynomial. Let's check the degree of what we have so far. $(x - 2)^4$ expands to a fourth-degree polynomial, and $(x - 4)$ is a first-degree polynomial. When we multiply them, the degrees add up: 4 + 1 = 5. Perfect! We're on the right track.

Adding the Leading Coefficient

We're not quite done yet! The problem also tells us that the leading coefficient must be 2. Right now, if we were to expand our polynomial, the leading coefficient would be 1 (because the highest power of $x$ in $(x - 2)^4$ is $x^4$, and when multiplied by $x$ from $(x - 4)$, we get $x^5$ with a coefficient of 1).

To fix this, we simply multiply the entire expression by 2. This will ensure that the coefficient of $x^5$ is 2, as required. So, our polynomial now looks like this:

$f(x) = 2(x - 2)^4(x - 4)$

And there we have it! This is the polynomial function that satisfies all the conditions given in the problem. It's a fifth-degree polynomial, has a zero of 2 with multiplicity 4, a zero of 4, and a leading coefficient of 2. Awesome!

Expanding the Polynomial (Optional)

Now, you could totally leave your answer as $f(x) = 2(x - 2)^4(x - 4)$. It's correct and clearly shows the zeros and their multiplicities. However, sometimes you might want to see the polynomial in its expanded form. It's a bit more work, but let's do it for fun!

First, we need to expand $(x - 2)^4$. This is where the binomial theorem or repeated multiplication comes in handy. Let's do it step by step:

$(x - 2)^2 = (x - 2)(x - 2) = x^2 - 4x + 4$

Now, square this result to get $(x - 2)^4$:

$(x^2 - 4x + 4)^2 = (x^2 - 4x + 4)(x^2 - 4x + 4)$

Expanding this (brace yourselves!), we get:

$x^4 - 8x^3 + 24x^2 - 32x + 16$

Okay, that's the first part done. Now we need to multiply this by $(x - 4)$:

$(x^4 - 8x^3 + 24x^2 - 32x + 16)(x - 4)$

More expanding! This gives us:

$x^5 - 12x^4 + 56x^3 - 128x^2 + 128x - 64$

Finally, we multiply the whole thing by 2 (remember that leading coefficient?):

$2(x^5 - 12x^4 + 56x^3 - 128x^2 + 128x - 64)$

So, the fully expanded form of our polynomial is:

$f(x) = 2x^5 - 24x^4 + 112x^3 - 256x^2 + 256x - 128$

Whoa! That's a mouthful, but it's the same polynomial as $f(x) = 2(x - 2)^4(x - 4)$, just written differently. See how much easier it is to identify the zeros and their multiplicities in the factored form?

Key Concepts Revisited

Let's recap what we've learned in this awesome polynomial-building adventure. This will really solidify the concepts in your mind.

Zeros and Factors

The heart of constructing a polynomial lies in understanding the relationship between zeros and factors. Remember, a zero of a polynomial is a value of $x$ that makes the polynomial equal to zero. If $x = c$ is a zero, then $(x - c)$ is a factor of the polynomial. This is super important! In our example, because 2 and 4 are zeros, $(x - 2)$ and $(x - 4)$ are factors.

Multiplicity

Multiplicity tells us how many times a particular factor appears in the polynomial. A zero with multiplicity $n$ means the corresponding factor appears $n$ times. For instance, 2 having a multiplicity of 4 means $(x - 2)$ appears four times, giving us $(x - 2)^4$. Multiplicity affects the behavior of the graph of the polynomial at the zero – a topic for another discussion, perhaps!

Degree and Leading Coefficient

The degree of the polynomial (the highest power of $x$) dictates the general shape and behavior of the polynomial. A fifth-degree polynomial, like ours, can have up to five zeros (counting multiplicities). The leading coefficient, the number in front of the highest power of $x$, scales the polynomial vertically. A leading coefficient of 2, like in our example, stretches the polynomial vertically compared to a polynomial with a leading coefficient of 1.

The Power of Factored Form

We talked about both the factored form ($f(x) = 2(x - 2)^4(x - 4)$) and the expanded form ($f(x) = 2x^5 - 24x^4 + 112x^3 - 256x^2 + 256x - 128$) of our polynomial. While both represent the same function, the factored form is incredibly useful. It immediately reveals the zeros and their multiplicities, which are crucial for understanding the polynomial's behavior and graph. The expanded form, while sometimes necessary, hides these key features.

Conclusion

So, guys, we've successfully constructed a polynomial function from a set of specific properties. We started with the zeros and their multiplicities, incorporated the leading coefficient, and even expanded the polynomial for kicks. The key takeaways here are understanding the relationships between zeros, factors, multiplicity, degree, and the leading coefficient. These concepts are fundamental in polynomial algebra and will serve you well in your mathematical adventures!

Remember, practice makes perfect. Try constructing polynomials with different properties. Play around with the zeros, multiplicities, and leading coefficients. The more you practice, the more comfortable you'll become with these concepts. Keep exploring the fascinating world of polynomials, and who knows what you'll discover next!