Problem 1 - Algebra 2 and Trigonometry

Question

Sharon said that if \(\mathrm{f}(x)\) is a polynomial function and \(\mathrm{f}(a)=0,\) then \(a\) is a root of the function. Do you agree with Sharon? Explain why or why not.

Step-by-Step Solution

Answer

Yes, Sharon's statement is correct; \(f(a) = 0\) means \(a\) is a root.

Step 1: Define a Root of a Polynomial

In polynomial algebra, a root of a polynomial function \(f(x)\) is any value \(a\) for which the polynomial evaluates to zero, i.e., \(f(a) = 0\). This definition aligns with the concept of finding the x-intercepts on the graph of \(f(x)\).

Step 2: Confirm Sharon's Statement

Given that Sharon stated \(f(a) = 0\) implies \(a\) is a root, according to the definition provided in Step 1, this statement is correct. \(a\) is indeed a root because it satisfies the condition \(f(a) = 0\).

Step 3: Understanding Polynomial Zeroes

When \(f(x)\) is a polynomial function, its roots or zeroes are the values that cause \(f(x)\) to become zero. These are crucial points as they often indicate where the graph of the polynomial touches or crosses the x-axis.

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression that involves a sum of powers of one or more variables multiplied by coefficients. These functions are fundamental in algebra and are expressed in the form:

• \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \)

Here, \(a_n, a_{n-1}, \ldots, a_0\) are constant coefficients, and \(n\) is a non-negative integer representing the degree of the polynomial. Each term consists of a combination of coefficients and variables raised to whole number powers. The degree of the polynomial is determined by the highest power of the variable present. In everyday mathematical problems, polynomial functions are used in modeling and solving equations where changes in one quantity depend on variations in another.
Understanding polynomial functions is crucial because they form the building blocks for more advanced topics in mathematics, such as calculus and differential equations.

X-intercepts

The x-intercepts of a polynomial function represent the points where the graph crosses or touches the x-axis. At these points, the value of the polynomial function is zero. In other words, the x-intercepts are the solutions to the equation:

• \( f(x) = 0 \)

This is equivalent to finding the roots or zeroes of the polynomial, which we'll discuss further in the next section. To find the x-intercepts of a polynomial function, we solve the equation \( f(x) = 0 \) by substituting different values of \(x\) until the equation holds true.
Graphically, a polynomial may have as many x-intercepts as its degree, though some x-intercepts may coincide, being repeated roots. These points are of great interest in graphing because they give key insights into the behavior of the polynomial function just by simple observation.

Zeroes of Polynomials

Zeroes of a polynomial function are synonymous with the roots of the function. They are the values of \(x\) that make the polynomial equation equal to zero: \( f(x) = 0 \). Finding the zeroes of a polynomial is a critical step in solving polynomial equations and is important in various aspects of mathematics and applied sciences.
The zeroes tell us not only where the graph crosses the x-axis but also inform us about the factorization of the polynomial. If a polynomial has a zero at \(x = a\), it can be factored by dividing it by \(x - a\). This is especially useful for simplifying complex polynomial expressions or solving polynomial equations.
It's important to note that a polynomial of degree \(n\) can have up to \(n\) zeroes, including complex and repeated zeroes. Each zero represents a potential intersection point with the x-axis, giving a visual picture of the function's behavior across different intervals.