Key takeaway: The multiplicity of a zero is the number of times its factor appears in a polynomial, and it tells whether the graph crosses the x-axis there or only touches and turns.
Grade 11 – Algebra 2A-APR.B.3Functions and Graphs
Definition of Multiplicity of a Zero
The multiplicity of a zero counts how many times its factor repeats in a factored polynomial: in \((x-1)^2\), the zero at \(x = 1\) has multiplicity 2.
That count predicts what the graph does at the x-axis. With an odd count, the curve passes through the axis. With an even count, it bounces off. A multiplicity of 1 crosses like a line, 2 touches like a parabola’s vertex, and 3 crosses while flattening like \(x^3\) at the origin. The multiplicities of all zeros add up to the degree.
Formula and notation
The multiplicity of a zero \(r\) is the exponent \(m\) on its factor:
\[ (x – r)^m \]
Odd \(m\): the graph crosses at \(x = r\). Even \(m\): the graph touches and turns around. Higher \(m\) also makes the graph flatter near \(r\).
Key facts
- Add all multiplicities to get the degree of the polynomial.
- Odd multiplicity means crossing; even multiplicity means touching.
- A multiplicity of 3 or more flattens the graph near the zero.
- Counting each repeat gives a polynomial of degree n exactly n zeros.
Multiplicity of a Zero example
Take \(f(x) = (x + 2)(x – 1)^2(x – 4)^3\). Find every zero, state how often its factor repeats, decide what the curve does at that point, and give the degree.
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\(x = -2\): factor \((x+2)^1\), multiplicity 1 (odd), so the graph crosses.
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\(x = 1\): factor \((x-1)^2\), multiplicity 2 (even), so the graph touches and turns.
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\(x = 4\): factor \((x-4)^3\), multiplicity 3 (odd), so the graph crosses with a flat spot. Degree: \(1 + 2 + 3 = 6\).
The zeros are -2 (1), 1 (2) and 4 (3), the graph touches only at \(x = 1\), and the degree is 6.
Common mistake
Watch out
Rae says \(f\) has only three zeros, so its degree must be 3.
Why it is wrong
Ignoring the multiplicity of a zero means counting distinct zeros only. Each zero is counted by its multiplicity: \(1 + 2 + 3 = 6\) zeros in all, so the degree is 6. The leading term is \(x^6\), and both tails go up.
Related terms and lessons
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