Question

WRITING The graph of the constant polynomial function \(f(x)=2\) is a line that does not have any \(x\)-intercepts. Does the function contradict the Fundamental Theorem of Algebra? Explain.

Short answer

No, the function does not contradict the Fundamental Theorem of Algebra. This is because the theorem applies to non-constant polynomial functions, while \(f(x)=2\) is a constant (degree 0) function.

Step-by-step solution

Understand the Function

The function given is \(f(x)=2\). This is a constant function. This means that the output value will not change regardless of the input. It is not dependent on \(x\). It is a polynomial of degree 0.

Understand the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This means that every polynomial function of degree greater than 0 must have at least one \(x\)-intercept, which are the roots or solutions of the equation.

Analyze the function with respect to the Theorem

In this case, since the function \(f(x)=2\) is a constant function i.e., it is a polynomial of degree 0, the rules of the Fundamental Theorem of Algebra do not apply to it. Hence, the lack of x-intercepts does not contradict the theorem.

Key concepts

constant polynomials

A constant polynomial is a type of polynomial where the output value remains the same regardless of the input. In simpler terms, it means that no matter what value you substitute for the variable \(x\), you will always get the same result. This happens because a constant polynomial does not actually involve the variable \(x\).

For instance, in the polynomial function \(f(x) = 2\), the graph of this function is a horizontal line across the \(y\)-axis at \(y = 2\). This line will never touch or cross the \(x\)-axis, which means it has no \(x\)-intercepts. Here are some key points:

• Constant polynomials are of degree 0.
• Their graphs are straight horizontal lines on the coordinate plane.
• They always have the same value, indicating that no matter the \(x\) value input, the output is constant.

Understanding constant polynomials provides a foundation for more complex functions and highlights the diversity within the category of polynomial functions.

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is a crucial concept in mathematics that addresses the roots of polynomial equations. It asserts that every non-constant single-variable polynomial with complex coefficients has at least one complex root. In simpler terms, this means:

• If a polynomial has a degree greater than 0, it must cross the \(x\)-axis at least once, indicating at least one root.
• These roots may be real numbers or complex numbers.
• For example, a quadratic polynomial (degree 2) will generally have two roots, which might be real or complex.

The theorem applies universally across polynomial functions but specifically excludes constant polynomials. Constant polynomials, having no variables, result in functions that are flat lines with no intercepts, making them exempt from this theorem's assertion about roots. Therefore, a constant function like \(f(x) = 2\) does not contradict the Fundamental Theorem of Algebra because it is not subject to its constraints.

degree of polynomial functions

The degree of a polynomial function is a key characteristic that defines its behavior and the nature of its graph. The degree is determined by the highest power of the variable \(x\) present in the polynomial. This concept can be broken down as follows:

• The degree indicates the number of root solutions a polynomial can have, considering multiplicity.
• It also gives an idea of the polynomial's shape on a graph: higher-degree polynomials have more complex curves.
• A constant polynomial, being degree 0, is represented by a flat horizontal line with no roots.

Knowing the degree of a polynomial helps predict the intercepts and the general shape of the graph. It assists in understanding how a polynomial behaves, offering crucial insight into solving polynomial equations. For instance, a polynomial function of degree \(n\) can have up to \(n\) roots, accounting for its full graphic representation.