Multiplicity of Polynomials

Front

What is the definition of a polynomial?

Back

A polynomial is a mathematical expression consisting of variables (or indeterminates) raised to non-negative integer powers and coefficients, combined using addition, subtraction, and multiplication.

Front

What does the term 'zeros of a polynomial' refer to?

Back

The zeros of a polynomial are the values of the variable that make the polynomial equal to zero.

Front

What is meant by the 'multiplicity' of a zero?

Back

Multiplicity refers to the number of times a particular zero is repeated as a root of the polynomial.

Front

How does the multiplicity of a zero affect the graph of a polynomial?

Back

The multiplicity of a zero affects the behavior of the graph at that zero: odd multiplicities cross the x-axis, while even multiplicities touch the x-axis and turn back.

Front

What is the relationship between the degree of a polynomial and its zeros?

Back

The degree of a polynomial is equal to the total number of its zeros, counting multiplicities.

Front

What is the significance of a turning point on the x-axis in relation to zero multiplicity?

Back

A turning point on the x-axis indicates that the zero has an even multiplicity.

Front

How can you determine the multiplicity of a zero from a polynomial's factored form?

Back

The multiplicity of a zero can be determined by the exponent of the factor corresponding to that zero in the polynomial's factored form.

Front

What effect do even multiplicities have on the graph of a polynomial at the corresponding zero?

Back

Even multiplicities cause the graph to touch the x-axis at the zero and then bounce back.

Front

What effect do odd multiplicities have on the graph of a polynomial at the corresponding zero?

Back

Odd multiplicities cause the graph to cross the x-axis at the zero.

Front

If a polynomial has a zero of -5 with a multiplicity of 3, what does this imply about the graph at x = -5?

Back

The graph will cross the x-axis at x = -5 and will have a point of inflection at that point.