Zeros of Polynomial Functions

Front

What are the zeros of a polynomial function?

Back

The zeros of a polynomial function are the values of x for which the function P(x) = 0.

Front

How do you find the zeros of the polynomial function P(x) = (x - 1)(x + 2)(x - 3)?

Back

Set each factor equal to zero: x - 1 = 0 (x = 1), x + 2 = 0 (x = -2), x - 3 = 0 (x = 3). The zeros are x = 1, x = -2, x = 3.

Front

What is the degree of a polynomial?

Back

The degree of a polynomial is the highest power of the variable in the polynomial expression.

Front

What does it mean if a polynomial has an even degree?

Back

If a polynomial has an even degree, its end behavior is the same on both sides; it either rises or falls on both ends.

Front

What does it mean if a polynomial has an odd degree?

Back

If a polynomial has an odd degree, its end behavior is different on both sides; it rises on one end and falls on the other.

Front

What is a leading coefficient?

Back

The leading coefficient is the coefficient of the term with the highest degree in a polynomial.

Front

How can you determine the number of real zeros of a polynomial function?

Back

You can determine the number of real zeros by analyzing the degree of the polynomial and using the Fundamental Theorem of Algebra.

Front

What is the Fundamental Theorem of Algebra?

Back

The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n roots (including complex roots) in the complex number system.

Front

What is the significance of the sign of the leading coefficient?

Back

The sign of the leading coefficient affects the end behavior of the polynomial function; a positive leading coefficient means the function rises to the right, while a negative leading coefficient means it falls to the right.

Front

How many zeros does the polynomial function P(x) = 3x^4 + 4x - 8 have?

Back

This polynomial has 4 zeros, as it is a degree 4 polynomial.