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(π‘₯) is a polynomial function, then solving the equation 𝑓(π‘₯) = 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

Example 1: A Simple Quadratic Polynomial

Let, f(x)=x2βˆ’4 To find the real zeroes:
Solve: f(x)=0 x2βˆ’4=0 x2=4 x=Β±2 Real Zeroes: x=βˆ’2x=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)=x3βˆ’6x2+11xβˆ’6 To find real zeroes:
Solve: x3βˆ’6x2+11xβˆ’6=0 We can factor it: f(x)=(xβˆ’1)(xβˆ’2)(xβˆ’3) Real zeroes: x=1,x=2,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)=(xβˆ’2)2 f(x)=x2βˆ’4x+4 Zero: π‘₯ = 2 (with multiplicity 2)
Graphical Behavior:
The graph touches the x-axis at π‘₯=2 and turns around; it doesn’t cross.

How to Find Real Zeroes

Summary

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 π‘Ÿ is a zero of a polynomial 𝑓(π‘₯), and it appears more than once as a solution when the polynomial is factored, then π‘Ÿ is said to have multiplicity π‘š, where π‘š is the number of times (π‘₯βˆ’π‘Ÿ) appears as a factor.

Definition:

Let, f(x)=(xβˆ’r)mβ‹…g(x) Where, g(r)β‰ 0
Then π‘Ÿ is a zero of multiplicity π‘š.
Importance in Graphing Polynomial Functions
The multiplicity of a zero tells us how the graph behaves at the x-intercept π‘₯=π‘Ÿ:

Examples

Example 1: Multiplicity 1 (Simple Root)
Let, f(x)=(xβˆ’2)(x+1)(xβˆ’4) Zeroes Graph behavior:
The graph crosses the x-axis at all three zeroes. Example 2: Even Multiplicity
Let, f(x)=(xβˆ’3)2 Zeroes Graph behavior:
The graph touches the x-axis at π‘₯=3 and turns around β€” it does not cross. Example 3: Odd Multiplicity Greater than 1
Let, f(x)=(x+2)3 Zeroes: Graph behavior:
The graph crosses the x-axis at π‘₯=βˆ’2, but with a flatter or cubic-like curve β€” not a sharp straight crossing.

Example: f(x)=(xβˆ’2)2(x+1) Zeroes:
β€” multiplicity 2 at x=2, multiplicity 1 at x=βˆ’1.
Graph behavior:
The graph touches the x-axis at π‘₯=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:

Interactive Graphs

βˆ’10βˆ’8βˆ’6βˆ’4βˆ’202468020406080100
f(x)ZeroesAaf(x) = (x + 1)(x - 1)xf(x)
Odd MultiplicityOdd Multiplicityβˆ’10βˆ’8βˆ’6βˆ’4βˆ’202468020406080100
f(x)ZeroesAaf(x) = (x - 3)^1 (x + 2)^1xf(x)