Features of Polynomials

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.

Front

How does the degree of a polynomial affect its end behavior?

Back

The degree of a polynomial determines the direction of the end behavior: even degrees lead to the same direction at both ends, while odd degrees lead to opposite directions.

Front

What is the significance of the leading coefficient in determining end behavior?

Back

The sign of the leading coefficient (positive or negative) affects the direction of the end behavior for polynomials of even degree.

Front

What is the standard form of a polynomial?

Back

The standard form of a polynomial is expressed as a sum of terms in descending order of degree, typically written as a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0.

Front

How do you find the roots of a polynomial function?

Back

The roots of a polynomial function are found by setting the polynomial equal to zero and solving for x.

Front

What does it mean for a root to have multiplicity?

Back

Multiplicity refers to the number of times a particular root appears in the factorization of a polynomial. A root with multiplicity greater than one indicates that the graph touches the x-axis at that point.

Front

What is the result of expanding (x + 2i)(x - 2i)?

Back

The result of expanding (x + 2i)(x - 2i) is x^2 + 4.

Front

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

Back

A polynomial of degree n can have up to n roots, counting multiplicities.

Front

What is the end behavior of a polynomial with an even degree and a positive leading coefficient?

Back

As x approaches -∞, f(x) approaches +∞; as x approaches +∞, f(x) approaches +∞.

Front

What is the end behavior of a polynomial with an odd degree and a negative leading coefficient?

Back

As x approaches -∞, f(x) approaches +∞; as x approaches +∞, f(x) approaches -∞.