Question

Write a polynomial function f of least degree that has a leading coefficient of 1 and the given zeros. \(0,5,-5+\sqrt{8}\)

Short answer

The polynomial function of least degree with the given zeros and a leading coefficient of 1 is \(f(x) = x^3 - 5x^2 + (5- \sqrt{8})x\).

Step-by-step solution

Determine the Factors from Roots

Formulate the factors from each root by setting each root equal to x in the format of (x - root). This gives us the factors (x - 0), (x - 5), and (x - (-5+\(\sqrt{8}\))) which simplifies to (x), (x - 5), and (x + 5 -\(\sqrt{8}\)).

Multiply the Factors to Form the Polynomial

To determine the polynomial, multiply the factors together. So the polynomial f(x) = (x) * (x - 5) * (x + 5 - \(\sqrt{8}\)).

Simplify the Polynomial

Expand and simplify the polynomial to get the final function. By multiplying the factors, f(x) = \(x^3 - 5x^2 + (5- \sqrt{8})x\).

Key concepts

Root Zero of Polynomial

Understanding the root zero of a polynomial is critical when constructing or analyzing polynomial functions. A root, or zero, of a polynomial is basically a value for the variable that makes the polynomial equal to zero. It is the x-intercept on a graph of the polynomial and signifies where the graph touches or crosses the x-axis.

For instance, in the given exercise, the zeros provided are 0, 5, and \( -5+\sqrt{8} \). Each zero corresponds to a factor of the polynomial; if \( x \) is a zero, then \( x - \text{zero} \) is a factor. Hence, a zero of 0 leads to a factor of \( (x - 0) \), or simply \( x \). This process is a first step in crafting a polynomial that possesses the specified roots and behaves according to those roots on a graph. By user-friendliness, highlighting the connection between roots and the factors they produce is essential, as this lays the groundwork for building the entire polynomial function.

The root zero also assists in determining the function's behavior at a particular point, affecting how the graph will turn or cross the axis, a concept that should not be underestimated as a foundational building block in understanding polynomials.

Leading Coefficient

The leading coefficient is a term that often causes confusion, but is quite simple once you get the hang of it. It is the coefficient of the first term when the polynomial is written in standard form, that is, in descending powers of \( x \). This coefficient can tell us a lot about the graph of the polynomial, such as the end behavior and the width of the graph.

Specifically, the leading coefficient in our exercise is given to be 1. Therefore, when constructing our polynomial function, the highest power of \( x \) should have a coefficient of 1. This is important because it will dictate the stretch and reflection of the graph. For example, if the leading coefficient were negative, the graph would reflect across the x-axis. In educational content, clarity on this concept is paramount because it affects all subsequent computations and the proper graphing of the polynomial function. Students should be aware that leading coefficients influence the graph's direction as \( x \) approaches infinity or negative infinity.

Simplify Polynomial

Once you have the factors from the zeros, the next step is to simplify the polynomial. This involves expanding the factors and combining like terms to write the polynomial in its simplest form, ideally in standard form starting with the highest power of \( x \) and descending to the constant term.

From our exercise, after applying the multiplication of the factors \( (x) * (x - 5) * (x + 5 - \sqrt{8}) \), it is essential to expand these to form a single polynomial expression that represents the function \( f(x) \). Simplification is a skill that makes larger polynomials more manageable and is especially useful in calculus, optimization problems, and when finding intercepts on a graph. It's more than mechanical calculation; it's an art of making the complex comprehensible, something any educational platform should strive to teach with excellence. The provided exercise's final simplified form of \( f(x) = x^3 - 5x^2 + (5 - \sqrt{8})x \) is the resulting polynomial that neatly encapsulates all the specified properties from the roots to the leading coefficient with all terms clearly presented.