Zeros of a polynomial function are the values of the variable that make the function equal to zero. In other words, they are the solutions to the equation $P(x) = 0$.
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The zeros of a polynomial function correspond to the x-intercepts of its graph.
A polynomial of degree $n$ can have at most $n$ real zeros.
Complex zeros of polynomials with real coefficients always occur in conjugate pairs.
The Factor Theorem states that if $x = c$ is a zero of the polynomial $P(x)$, then $(x - c)$ is a factor of $P(x)$.
Multiplicity refers to the number of times a particular zero appears; if a zero has even multiplicity, the graph touches and turns around at this point, while an odd multiplicity means it crosses through.
Review Questions
What is the maximum number of real zeros for a polynomial function with degree 4?
Explain how you can determine whether a given value is a zero of a polynomial function.
Describe how complex zeros appear in polynomials with real coefficients.
Related terms
Factor Theorem: A theorem stating that $(x - c)$ is a factor of the polynomial $P(x)$ if and only if $P(c) = 0$.
Multiplicity: The number of times a particular zero appears in the factored form of a polynomial.
Complex Conjugate: If $a + bi$ is a complex zero, then its complex conjugate is $a - bi$. They appear in pairs for polynomials with real coefficients.