Zeros & Multiplicity

Front

What does it mean if the graph of a function crosses the x-axis?

Back

The zero has an odd multiplicity.

Front

How does a multiplicity of 2 affect the graph of a polynomial function?

Back

The graph will bounce off the x-axis at that zero.

Front

Write a polynomial function with zeros at 5 (multiplicity 2), 3 (multiplicity 1), and 0 (multiplicity 4).

Back

Front

Identify the real zeros and their multiplicities for the function f(x) = x^2(x - 2)^3(x - 7).

Back

0 (Multiplicity 2), 2 (Multiplicity 3), 7 (Multiplicity 1).

Front

What effect does a multiplicity of 1 have on the graph of a polynomial function?

Back

The graph will cross the x-axis at that zero.

Front

Define 'zero' in the context of polynomial functions.

Back

A zero of a polynomial function is a value of x that makes the function equal to zero.

Front

What is the significance of multiplicity in polynomial functions?

Back

Multiplicity indicates how many times a zero is repeated and affects the graph's behavior at that zero.

Front

How does a zero with an even multiplicity behave at the x-axis?

Back

The graph touches the x-axis but does not cross it.

Front

What is the general form of a polynomial function with given zeros?

Back

If a polynomial has zeros at x = a, x = b, and x = c with multiplicities m, n, and p, it can be expressed as f(x) = k(x - a)^m(x - b)^n(x - c)^p, where k is a constant.

Front

Explain the term 'multiplicity' in relation to polynomial zeros.

Back

Multiplicity refers to the number of times a particular zero appears in the factorization of a polynomial.