Real Zeroes and Their Multiplicities

Real Zeroes of Polynomial Functions

The greal zeroes (or real roots ) of a polynomial function are the values of \( x \) for which the polynomial equals zero. In other words, a real zero is a real number \( x = r \) such that \[ f(r) = 0 \].

If 𝑓(𝑥) is a polynomial function, then solving the equation 𝑓(𝑥) = 0 gives us the zeroes of the polynomial. If the solution(s) are real numbers, they are called real zeroes. It's the point where the graph touches or crosses the x-axis.

Why Real Zeroes Matter

• Real zeroes tell us where the graph of the polynomial crosses or touches the x-axis.
• If 𝑥 = 𝑟 is a real zero of a polynomial 𝑓 ( 𝑥 ), then ( 𝑟 , 0 ) is an x-intercept of the graph of 𝑓.
• Real zeroes help us understand the behavior and shape of the graph.

Example 1: A Simple Quadratic Polynomial

Let, \[f(x) = x^2 - 4\] To find the real zeroes:
Solve: \[f(x) = 0 \] \[ x^2 - 4 = 0\] \[x^2 = 4\] \[x = \pm 2\] Real Zeroes: \[x = - 2 \quad x = 2\] Graphical Interpretation:
The graph of \(f(x) = 0 \) is a parabola that crosses the x-axis at (-2,0) and (2,0).

Example 2: A Cubic Polynomial

Let, \[f(x) = x^ 3 - 6x^2 + 11 x - 6 \] To find real zeroes:
Solve: \[ x^ 3 - 6x^2 + 11 x - 6 = 0\] We can factor it: \[ f(x) = (x - 1)(x - 2)(x - 3) \] Real zeroes: \[x = 1, \quad x= 2, \quad x = 3\] Graphical Interpretation:
The graph of \(f(x) = 0 \) crosses the x-axis at three points (1,0), (2,0), and (3,0).

Types of Real Zeroes:

1. Single Zero: The graph crosses the x-axis at that point (odd multiplicity).
2. Repeated Zero: The graph touches the x-axis and turns around (even multiplicity).

Example 3: Repeated Zero

Let, \[f(x) = (x - 2)^2 \] \[f(x) = x^2 - 4x + 4 \] Zero: 𝑥 = 2 (with multiplicity 2)
Graphical Behavior:
The graph touches the x-axis at 𝑥=2 and turns around; it doesn’t cross.

How to Find Real Zeroes

• Factor the polynomial, if possible
• Use the Rational Root Theorem or synthetic division to test for zeroes.
• Use the quadratic formula, if the polynomical is quadratic.
• Use graphing tools to estimate or visualize the zeroes.

Summary

• Real zeroes of a polynomial are the real values of 𝑥 where 𝑓 (𝑥) = 0.
• They represent the x-intercepts of the graph.
• You can find them through factoring, using formulas, or graphing.
• Understanding real zeroes helps you analyze and sketch polynomial functions.

Multiplicity of Real Zeroes

The multiplicity of a real zero of a polynomial function refers to the number of times a particular zero repeats as a root of the equation.
In simple terms:
Multiplicity means how many times a zero is repeated, that is, If a real number 𝑟 is a zero of a polynomial 𝑓(𝑥), and it appears more than once as a solution when the polynomial is factored, then 𝑟 is said to have multiplicity 𝑚, where 𝑚 is the number of times (𝑥−𝑟) appears as a factor.

Definition:

Let, \[f(x) = (x - r)^m \cdot g(x)\] Where, \(g(r) \ne 0 \)
Then 𝑟 is a zero of multiplicity 𝑚.
Importance in Graphing Polynomial Functions
The multiplicity of a zero tells us how the graph behaves at the x-intercept 𝑥=𝑟:

• Odd multiplicity (e.g., 1, 3, 5...):
  The graph crosses the x-axis at 𝑥= 𝑟.
• Even multiplicity (e.g., 2, 4, 6...):
  The graph touches the x-axis at 𝑥=𝑟 and turns around (does not cross).

Examples

Example 1: Multiplicity 1 (Simple Root)
Let, \[ f(x) = (x - 2)(x + 1)(x - 4) \] Zeroes

• x = 2 (multiplicity 1)
• x = - 1 (multiplicity 1)
• x = 4 (multiplicity 1)

Graph behavior:
The graph crosses the x-axis at all three zeroes. Example 2: Even Multiplicity
Let, \[f(x) = (x - 3)^2 \] Zeroes

• x=3, multiplicity 2

Graph behavior:
The graph touches the x-axis at 𝑥=3 and turns around — it does not cross. Example 3: Odd Multiplicity Greater than 1
Let, \[f(x) = (x + 2)^3 \] Zeroes:

• x=−2, multiplicity 3

Graph behavior:
The graph crosses the x-axis at 𝑥=−2, but with a flatter or cubic-like curve — not a sharp straight crossing.

Example: \[ f(x) = (x - 2)^2(x + 1) \] Zeroes:
— multiplicity 2 at \( x = 2 \), multiplicity 1 at \( x = -1 \).
Graph behavior:
The graph touches the x-axis at 𝑥=2, but crosses at x = -1.

Graphical Summary

Multiplicity	Behavior at x-intercept
1	Crosses the x-axis sharply
Even (2, 4)	Touches and turns (bounces back)
Odd > 1 (3,5)	Crosses but flattens at the root

Why Multiplicity is Important

1. Shapes the graph: Tells whether the graph crosses or bounces at x-intercepts.
2. Affects curve behavior: High multiplicity makes the graph flatter at the intercept.
3. Helps in sketching: Knowing multiplicities helps sketch an accurate graph without full plotting.

Real-World Analogy

Imagine the x-axis is a floor:

• A zero with multiplicity 1 is like walking across the floor.
• A zero with even multiplicity is like bumping into the floor and bouncing back.
• A zero with odd multiplicity greater than 1 is like gently sliding over the floor while crossing.