Multiplicity of Zeros in Polynomials

When a polynomial function is written in factored form, each zero (or root) comes from a factor of the form \((x - a)\). The multiplicity of that zero is simply how many times the factor \((x - a)\) is repeated in the factorization.

For example, in \(f(x) = (x - 2)^3(x + 1)\) the zero \(x = 2\) has multiplicity \(3\), because \((x-2)\) appears three times, and the zero \(x = -1\) has multiplicity \(1\), because \((x+1)\) appears only once. Multiplicity is closely tied to the Factor theorem, which tells us that every zero of a polynomial corresponds to a factor of that polynomial, and vice versa.

If a polynomial is already factored, finding multiplicity is just counting repeated factors.

Example: List each real zero of \(g(x) = (x-1)^2(x+3)(x-5)^4\) and its multiplicity.

• \(x = 1\): multiplicity \(2\)
• \(x = -3\): multiplicity \(1\)
• \(x = 5\): multiplicity \(4\)

Notice the exponents add up to \(2 + 1 + 4 = 7\), which matches the degree of \(g(x)\). This is always true: the sum of the multiplicities of all real and complex zeros equals the degree of the polynomial.

If you are only given the expanded form, you first need to factor the polynomial to see the repeated roots. Tools like the rational zero test and synthetic division can help locate zeros, and repeating the same zero through division reveals its multiplicity. Once you know the zeros and their multiplicities, you can also work backwards and build the equation, which is the idea behind determining the equation of a polynomial function from its zeros.

Multiplicity is not just a counting exercise; it controls exactly how the graph behaves near each x-intercept.

• Odd multiplicity (1, 3, 5, ...): the graph crosses the x-axis at that zero.
• Even multiplicity (2, 4, 6, ...): the graph touches the x-axis and turns back, without crossing.
• The higher the multiplicity, the flatter the graph looks as it approaches that zero.

Consider \(f(x) = (x-1)^2(x+2)\). The zero at \(x = 1\) has multiplicity \(2\) (even), so the graph touches the axis and bounces back up. The zero at \(x = -2\) has multiplicity \(1\) (odd), so the graph crosses straight through.

Now compare that to \(h(x) = (x+1)^3(x-2)\). The zero at \(x = -1\) has multiplicity \(3\) (odd), so the graph still crosses the axis, but it flattens out and almost pauses as it does so, unlike a simple straight-through crossing at a multiplicity-1 zero such as \(x = 2\).

Recognizing multiplicity lets you sketch a polynomial's graph quickly just from its factored form, without plotting many points. It also helps when solving equations, since a repeated root only needs to be listed once even though it is counted more than once toward the polynomial's degree. This skill shows up again when you study applications of polynomial functions and when using Descartes' rule of signs to predict the number of positive and negative real zeros.

Given \(p(x) = -(x+4)(x-1)^2(x-3)^3\):

• \(x = -4\): multiplicity \(1\), odd, graph crosses the axis.
• \(x = 1\): multiplicity \(2\), even, graph touches and bounces.
• \(x = 3\): multiplicity \(3\), odd, graph crosses but flattens.

Degree check: \(1 + 2 + 3 = 6\), matching the degree of \(p(x)\).

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