4-13 Polynomial Functions with Multiplicity HW

Front

What is a polynomial function?

Back

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n ≠ 0.

Front

What does the term 'zero' refer to in polynomial functions?

Back

A zero of a polynomial function is a value of x for which the function evaluates to zero, meaning the graph of the function intersects the x-axis at that point.

Front

What is multiplicity in the context of polynomial functions?

Back

Multiplicity refers to the number of times a particular zero occurs in a polynomial function. It indicates how many times the graph touches or crosses the x-axis at that zero.

Front

How does a zero with odd multiplicity behave on a graph?

Back

A zero with odd multiplicity crosses the x-axis at that point, indicating a change in sign of the function.

Front

How does a zero with even multiplicity behave on a graph?

Back

A zero with even multiplicity touches the x-axis but does not cross it, resulting in a 'bounce' effect.

Front

What is the significance of the degree of a polynomial function?

Back

The degree of a polynomial function indicates the highest power of the variable in the function. It determines the number of zeros and the end behavior of the graph.

Front

How can you determine the number of real zeros from a polynomial function's graph?

Back

The number of real zeros can be determined by counting the points where the graph intersects the x-axis.

Front

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

Back

The degree of a polynomial function is equal to the maximum number of real zeros it can have, counting multiplicities.

Front

What does it mean if a polynomial has a zero at x = 1 with multiplicity 2?

Back

It means that the graph of the polynomial touches the x-axis at x = 1 and bounces back, indicating that the zero is repeated.

Front

If a polynomial has a zero at x = -4 with multiplicity 1, what does this imply?

Back

It implies that the graph crosses the x-axis at x = -4, indicating a change in sign of the function.