Zeros

Definition

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 $P(x) = 0$.

5 Must Know Facts For Your Next Test

1. The zeros of a polynomial function correspond to the x-intercepts of its graph.
2. A polynomial of degree $n$ can have at most $n$ real zeros.
3. Complex zeros of polynomials with real coefficients always occur in conjugate pairs.
4. The Factor Theorem states that if $x = c$ is a zero of the polynomial $P(x)$, then $(x - c)$ is a factor of $P(x)$.
5. 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.