The zeroes of a polynomial are critical in understanding its behavior and characteristics. In this section, we explore the geometrical interpretation of these zeroes on the Cartesian plane.

1. Definition of Zeroes of a Polynomial

A polynomial $P(x)$ is a mathematical expression of the form:

$$P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$$

where $a_n, a_{n-1}, \ldots, a_1, a_0$ are constants, and $n$ is a non-negative integer. The values of $x$ for which $P(x) = 0$ are called the zeroes of the polynomial.

2. Geometrical Representation

When we graph the polynomial $P(x)$ on a Cartesian coordinate system, the zeroes correspond to the points where the graph intersects the $x$-axis. This intersection indicates that the output (or $y$-value) of the polynomial is zero for those specific $x$-values.

3. Examples

Consider the polynomial:

$$P(x) = x^2 - 5x + 6$$

To find the zeroes, we set the polynomial equal to zero:

$$x^2 - 5x + 6 = 0$$

Factoring gives:

$$ (x - 2)(x - 3) = 0 $$

The zeroes are $x = 2$ and $x = 3$. Graphing this polynomial will show that it intersects the $x$-axis at the points $(2, 0)$ and $(3, 0)$.

4. Degree of the Polynomial

The degree of the polynomial plays a significant role in determining the number of zeroes and their multiplicity:

• A polynomial of degree $n$ can have up to $n$ zeroes.
• Zeroes may be real or complex, and their multiplicity indicates how many times a particular zero is repeated.

5. Behavior of the Polynomial at Zeroes

The behavior of the polynomial near its zeroes can be understood as follows:

• If a zero has odd multiplicity, the graph will cross the $x$-axis at that zero.
• If a zero has even multiplicity, the graph will touch the $x$-axis but not cross it at that zero.

6. Graphical Illustrations

To visualize these concepts, consider the following cases:

• Example 1: For $P(x) = (x - 1)(x - 2)$, the graph intersects the $x$-axis at $(1, 0)$ and $(2, 0)$, indicating that both zeroes are of odd multiplicity.
• Example 2: For $P(x) = (x - 1)^2$, the graph touches the $x$-axis at $(1, 0)$ and does not cross it, indicating that the zero at $x = 1$ has even multiplicity.

7. Importance of Zeroes

The zeroes of a polynomial have significant implications in various fields, including:

• Root Finding: Zeroes provide critical values for solving equations.
• Optimization: They help in determining maximum and minimum values of functions.
• Graphing: Knowing the zeroes aids in sketching the polynomial's graph accurately.

8. Summary of Key Points

• The zeroes of a polynomial represent the $x$-intercepts of its graph.
• The degree of the polynomial indicates the maximum number of zeroes.
• Understanding the behavior of the graph at zeroes helps in identifying their multiplicity.

9. Practice Problems

To reinforce your understanding, try finding the zeroes of the following polynomials and sketch their graphs:

• 1. $P(x) = x^3 - 6x^2 + 9x$
• 2. $P(x) = x^2 - 4$
• 3. $P(x) = x^4 - 1$

By engaging with these problems, you will deepen your comprehension of the geometrical meaning of the zeroes of polynomials.