Multiplicity of a Zero
In short: Multiplicity counts how many times a factor repeats in a polynomial, and that count decides whether the graph crosses the x-axis or only touches and turns.
Algebra 2Functions and GraphsA-APR.B.3
What is Multiplicity of a Zero?
When a factor such as \((x – 3)\) appears more than once in a factored polynomial, its exponent is the multiplicity of the zero \(x = 3\): the number of times that zero is repeated.
The exponent controls the picture. An odd exponent means the graph crosses the axis there, and 3 or more also flattens it while passing. An even exponent means the graph touches the axis and turns back. Adding all the exponents gives the degree of the polynomial.
How to spot it
- A polynomial is given already factored, with some factors raised to powers such as 2 or 3.
- A graph bounces off the x-axis at a point, or flattens out while it crosses.
- A question asks how many zeros a polynomial has when repeats are counted.
- You must build an equation from a sketch showing where the curve crosses and where it touches.
The method
- Set each factor equal to zero to list the zeros.
- Read the exponent on each factor; that exponent is the multiplicity of its zero.
- Odd exponent: crossing. Even exponent: touching and turning. Exponent 3 or more: flatter near the axis.
- Add the exponents to find the degree, and use the leading term to describe the ends.
Multiplicity of a Zero: solved example
Problem
For \(f(x) = x(x – 3)^2(x + 1)^3\), list each zero with its repeat count, describe what the graph does there, and find the degree.
Solution
\(x = 0\) appears once: the graph crosses. \(x = 3\) appears twice: the graph touches the axis and turns. \(x = -1\) appears three times: the graph flattens and crosses.
Add the exponents: \(1 + 2 + 3 = 6\), so the degree is 6. The leading coefficient is positive and the degree is even, so both ends rise.
Try it
Question 1
In \(g(x) = (x – 2)^4(x + 5)\), give the repeat count of each zero and the degree.
Show the answer
The zero 2 repeats four times, so the graph touches and turns; \(-5\) appears once and the graph crosses. The degree is \(4 + 1 = 5\).
Question 2
A cubic polynomial touches the x-axis at 1 and crosses it at \(-4\). Write a possible equation.
Show the answer
The touching zero needs an even exponent: \(f(x) = (x – 1)^2(x + 4)\), which has degree 3.
Multiplicity of a Zero vs. Zeros of a Function
A zero is a location, an x-value where the output is 0. Multiplicity is a separate fact about that location. In \((x – 2)^4(x + 5)\) there are two distinct zeros, 2 and \(-5\), but five zeros when repeats are included: 2 four times and \(-5\) once.
Keep going
Lessons
Worksheets and quizzes
- Zeros and Graphs of Polynomial Functions – Worksheet
- Zeros and Graphs of Polynomial Functions – Quiz
Related words
Whole dictionary: A–Z · Algebra 2 · Functions and Graphs