HAA Zeros and End Behavior

Front

What are the zeros of the function f(x) = (x+2)(x-3)²(x-4)?

Back

The zeros are -2, 3 (multiplicity 2), and 4.

Front

What does the term 'multiplicity' mean in relation to polynomial zeros?

Back

Multiplicity refers to the number of times a particular zero appears as a root of the polynomial. For example, if a zero has a multiplicity of 2, it means that the factor corresponding to that zero is squared.

Front

According to the Fundamental Theorem of Algebra, how many zeros does a polynomial function of degree n have?

Back

A polynomial function of degree n has exactly n zeros, counting multiplicities.

Front

What is the end behavior of a polynomial function?

Back

The end behavior of a polynomial function describes how the function behaves as x approaches positive or negative infinity. It is determined by the leading term of the polynomial.

Front

How do you determine the end behavior of the function f(x) = - (x² - 9)(x + 4)?

Back

Since the leading coefficient is negative and the degree is odd, as x approaches positive infinity, f(x) approaches negative infinity, and as x approaches negative infinity, f(x) approaches positive infinity.

Front

What is the significance of the leading coefficient in a polynomial function?

Back

The leading coefficient determines the direction of the end behavior of the polynomial function. If it is positive, the ends of the graph will rise in both directions; if negative, the ends will fall.

Front

What are the zeros of the function f(x) = -7x(x-5)³(x+6)?

Back

The zeros are x = 0, x = 5 (multiplicity 3), and x = -6.

Front

How can you identify the zeros of a polynomial function from its factored form?

Back

The zeros can be identified by setting each factor equal to zero and solving for x.

Front

What is the relationship between the degree of a polynomial and the number of zeros it can have?

Back

The degree of a polynomial indicates the maximum number of zeros it can have, including complex and repeated zeros.

Front

What does it mean for a polynomial to have a zero of multiplicity 3?

Back

A zero of multiplicity 3 means that the factor corresponding to that zero appears three times in the polynomial, indicating that the graph will touch the x-axis at that point and turn around.