Finding A Degree 3 Polynomial: Zeros, Coefficients, & Solutions

Hey math enthusiasts! Let's dive into a cool polynomial problem. We're tasked with crafting a degree 3 polynomial, meaning the highest power of 'x' will be 3. This polynomial needs to have some specific features: integer coefficients (whole numbers) and particular zeros (where the polynomial equals zero). The zeros we're working with are -2/5, 5/4, and 3. Don't worry; it might sound complex, but we'll break it down step by step. By the end of this article, you'll be a pro at constructing polynomials with given zeros and understanding their properties. This is a fundamental skill in algebra, used in various fields like engineering, physics, and computer science. So, let's get started and make this polynomial magic happen!

Understanding the Basics: What are Polynomials and Zeros?

Alright, before we jump in, let's quickly recap some basics. A polynomial is an expression with variables and coefficients. For instance, f(x) = 3x^2 + 2x - 1 is a polynomial. The degree of the polynomial is the highest power of the variable (in this case, 2). Zeros of a polynomial are the x-values that make the polynomial equal to zero. They are essentially the points where the graph of the polynomial crosses the x-axis. Finding zeros is crucial because it tells us where the function behaves in specific ways. Also, integer coefficients mean the numbers in front of the x terms and the constant term are all integers. In our case, the zeros are -2/5, 5/4, and 3. These are the x-values where the polynomial will equal zero.

Think of it like this: if you plug in any of these zeros into our polynomial equation, the entire equation will collapse to zero. This relationship between zeros and polynomials is key to solving our problem. Every polynomial can be represented as a product of linear factors. For each zero, there's a corresponding linear factor. This fundamental concept is the foundation of how we construct our polynomial from its zeros. Let's see how this principle unfolds as we delve into the process of constructing our degree 3 polynomial with the given zeros.

Constructing the Polynomial: From Zeros to Factors

Now, the fun part! We know our zeros: -2/5, 5/4, and 3. From these, we can create the factors of the polynomial. Remember that if r is a zero of a polynomial, then (x - r) is a factor. So, for each of our zeros, we get a factor:

• For -2/5, the factor is (x - (-2/5)) = (x + 2/5).
• For 5/4, the factor is (x - 5/4).
• For 3, the factor is (x - 3).

To construct the polynomial, we multiply these factors together. However, we have a slight problem: the first two factors have fractions, and we need integer coefficients. To fix this, we can manipulate the fractional factors to get rid of the fractions. Let's handle the first factor, (x + 2/5). To eliminate the fraction, multiply the entire factor by 5. This gives us 5(x + 2/5) = 5x + 2. Similarly, for the second factor, (x - 5/4), multiply by 4 to get rid of the fraction: 4(x - 5/4) = 4x - 5. Now, our polynomial is the product of these adjusted factors and the third factor: f(x) = (5x + 2)(4x - 5)(x - 3). Now we've successfully transformed the zeros into factors and have the building blocks of our polynomial. It's like having all the ingredients ready to bake a cake. The next step is to combine these components to form our complete polynomial function. Let's proceed to the next step, where we'll expand these factors and simplify the expression.

Expanding and Simplifying: Bringing it All Together

We have our factors, and now it's time to expand and simplify the expression to find the final form of our polynomial. We start by multiplying the first two factors: (5x + 2)(4x - 5). Using the distributive property (or the FOIL method), we get:

20x^2 - 25x + 8x - 10 = 20x^2 - 17x - 10

Now, we multiply this result by the third factor, (x - 3):

(20x^2 - 17x - 10)(x - 3)

Again, using the distributive property:

20x^3 - 60x^2 - 17x^2 + 51x - 10x + 30

Combine like terms:

20x^3 - 77x^2 + 41x + 30

So, the polynomial f(x) with integer coefficients and zeros of -2/5, 5/4, and 3 is f(x) = 20x^3 - 77x^2 + 41x + 30. This is our final answer! We've successfully constructed the polynomial by converting the zeros into factors, adjusting for integer coefficients, and expanding the factors to obtain the standard form of the polynomial. This process exemplifies the direct connection between the zeros of a polynomial and its factored and expanded forms.

Verifying the Solution: Checking the Zeros

Let's do a quick check to make sure our polynomial works. Although we constructed the polynomial using the zeros, we can verify that it does have the specified zeros. Unfortunately, directly plugging in the fractional zeros into the expanded form is cumbersome. However, we can confirm that the original zeros were indeed the basis of our polynomial's structure. Because we know that (5x + 2) corresponds to the zero -2/5, (4x - 5) corresponds to the zero 5/4, and (x - 3) corresponds to the zero 3, we can confirm our solution. In this context, verification confirms that the polynomial aligns with the problem's original requirements.

Another method to verify involves graphing the polynomial and visually confirming that the graph intersects the x-axis at the zeros we started with. This graphical confirmation is a great way to visualize the solution and reinforce the concepts we've covered. Furthermore, if the graph of f(x) = 20x^3 - 77x^2 + 41x + 30 crosses the x-axis at approximately -0.4, 1.25, and 3, then our solution is correct. The intersection points represent the zeros, where the function's value equals zero.

Conclusion: Polynomials Demystified

There you have it! We've successfully constructed a degree 3 polynomial with integer coefficients and the given zeros of -2/5, 5/4, and 3. We started with the zeros, converted them into factors, adjusted for integer coefficients, expanded the factors, and simplified to get the standard form of the polynomial. Remember, this process is a fundamental skill in algebra, and understanding how to work with zeros and factors is crucial for advanced topics. This skill opens the door to understanding more complex polynomial behaviors and their applications in various mathematical and scientific fields.

I hope you found this explanation helpful. Now, you can tackle similar problems with confidence. Keep practicing, and you'll become a polynomial master in no time! If you have any questions or want to explore more problems, feel free to ask. Happy solving!