Finding A Polynomial Of Degree 3 With Zeros 8, -3, And 0

In the realm of polynomial functions, finding a polynomial that satisfies specific conditions, such as having certain zeros, is a fundamental problem. This article delves into the process of constructing a polynomial f(x) of degree 3, given that its zeros are 8, -3, and 0. We will explore the relationship between zeros and factors of a polynomial, and subsequently, express the polynomial in its factored form. This exploration will not only reinforce core concepts in algebra but also demonstrate the practical application of these concepts in constructing mathematical models.

Understanding the relationship between the zeros of a polynomial and its factors is paramount in this endeavor. A zero of a polynomial, denoted as x = a, implies that the polynomial evaluates to zero when x is substituted with a. This, in turn, means that (x - a) is a factor of the polynomial. For a polynomial of degree 3, we anticipate having at most three zeros, and consequently, three linear factors. The factored form of a polynomial provides a concise representation that directly reveals its zeros, making it a valuable tool in polynomial analysis and manipulation. The journey to find the polynomial f(x) involves leveraging the given zeros to construct the corresponding factors and then combining these factors to form the polynomial. The subsequent sections will meticulously walk through this process, offering insights into the underlying mathematical principles and techniques.

The core concept to grasp here is the intimate relationship between the zeros of a polynomial and its factors. A zero of a polynomial f(x) is a value x = a such that f(a) = 0. In simpler terms, it's the x-value where the polynomial's graph intersects the x-axis. This concept is deeply connected to the factor theorem, which states that if a is a zero of f(x), then (x - a) is a factor of f(x). Conversely, if (x - a) is a factor of f(x), then a is a zero of f(x). This bidirectional relationship forms the cornerstone of our approach to constructing the polynomial.

To further clarify, consider a polynomial like f(x) = (x - 2)(x + 1). The zeros of this polynomial are x = 2 and x = -1, because substituting these values into the polynomial makes it equal to zero. The factors are (x - 2) and (x + 1), which are directly derived from the zeros. This simple example illustrates the fundamental connection we'll utilize to solve our problem. Given the zeros 8, -3, and 0, we can immediately identify the corresponding factors as (x - 8), (x + 3), and (x - 0), respectively. The factor (x - 0) simplifies to just x, making our task even more straightforward. This understanding is crucial because it allows us to translate the information about zeros into a tangible algebraic form – the factors – which we can then manipulate to construct the polynomial. The next step involves combining these factors to form the polynomial f(x), ensuring that it has the specified degree and zeros.

Now that we understand the link between zeros and factors, we can embark on the construction of our polynomial f(x). We are given three zeros: 8, -3, and 0. Following the principle we discussed, these zeros translate directly into the following factors:

• Zero 8 corresponds to the factor (x - 8)
• Zero -3 corresponds to the factor (x + 3)
• Zero 0 corresponds to the factor x (since x - 0 = x)

To form a polynomial with these zeros, we simply multiply these factors together. This gives us a preliminary factored form: f(x) = a x(x - 8)(x + 3), where a is a leading coefficient. The inclusion of a is crucial because multiplying by a constant does not alter the zeros of the polynomial but does affect its overall shape and vertical stretch or compression. The problem specifies that we need a polynomial of degree 3. Our current factored form, x(x - 8)(x + 3), when expanded, will indeed result in a cubic polynomial (a polynomial of degree 3). To see this, imagine multiplying the x term by the other two factors; you'll get a term with x cubed, which is the highest power of x, thus confirming the degree.

Therefore, we can express our polynomial in the factored form f(x) = a x(x - 8)(x + 3)*. The final touch is to determine the value of a. If no other conditions are given, we can assume the simplest case where a = 1. This yields our polynomial in factored form: f(x) = x(x - 8)(x + 3). This is the most concise representation that directly reveals the zeros of the polynomial. It satisfies the condition of having degree 3 and possessing the specified zeros. In scenarios where additional information is provided, such as a specific point the polynomial must pass through, we can use that information to solve for a unique value of a. However, in the absence of such constraints, a = 1 provides the simplest and most common solution. The next step, though not explicitly required by the problem, would be to expand this factored form to obtain the polynomial in standard form, which is a sum of terms with decreasing powers of x. This expanded form provides an alternative representation of the same polynomial and can be useful in different contexts.

While the problem specifically asks for the polynomial in factored form, expanding it to its standard form can provide a deeper understanding of its structure and behavior. It also serves as a useful exercise to reinforce algebraic manipulation skills. Starting with our factored form, f(x) = x(x - 8)(x + 3), we proceed with the expansion step by step.

First, let's multiply the two binomial factors, (x - 8) and (x + 3). Using the distributive property (often remembered by the acronym FOIL – First, Outer, Inner, Last), we get:

(x - 8)(x + 3) = xx + x3 - 8x - 83 = x^2 + 3x - 8x - 24

Simplifying this expression by combining like terms (the x terms), we obtain:

x^2 - 5x - 24

Now, we substitute this back into our expression for f(x), giving us:

f(x) = x(x^2 - 5x - 24)

Finally, we distribute the x term across the trinomial:

f(x) = xx^2 - x5x - x24 = x^3 - 5x^2 - 24x*

Therefore, the expanded form of our polynomial is f(x) = x^3 - 5x^2 - 24x. This is a cubic polynomial, as expected, and it represents the same function as the factored form f(x) = x(x - 8)(x + 3). The standard form clearly shows the coefficients of each term, which can be useful for certain analyses, such as determining the polynomial's end behavior or applying the rational root theorem. However, the factored form is more directly informative about the polynomial's zeros. Both forms have their advantages and are valuable in different contexts. This expansion process demonstrates how we can move between these two representations, highlighting the flexibility and power of algebraic manipulation.

In this article, we successfully found a polynomial f(x) of degree 3 with the given zeros 8, -3, and 0. We began by understanding the fundamental relationship between the zeros of a polynomial and its factors. We translated each zero into its corresponding factor and then combined these factors to form the polynomial in factored form. The resulting polynomial is f(x) = x(x - 8)(x + 3). This factored form provides a concise and direct representation of the polynomial, clearly displaying its zeros.

We also explored the optional step of expanding the factored form to obtain the polynomial in standard form, f(x) = x^3 - 5x^2 - 24x. This exercise demonstrated the equivalence between the two forms and highlighted the different insights each provides. The factored form is particularly useful for identifying zeros, while the standard form can be advantageous for other types of analysis.

This process underscores the importance of understanding the interplay between zeros, factors, and the various forms of representing polynomials. The ability to construct polynomials with specific properties is a valuable skill in mathematics, with applications spanning from curve fitting and modeling to more advanced concepts in calculus and differential equations. The problem-solving approach we employed can be generalized to find polynomials with any given set of zeros and a specified degree, making it a versatile tool in the realm of polynomial algebra. Mastering these techniques opens the door to a deeper understanding of polynomial functions and their diverse applications in mathematics and beyond.