Polynomial Functions

Front

What is a polynomial?

Back

A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to non-negative integer powers and coefficients. For example, f(x) = 2x^3 - 4x + 7 is a polynomial.

Front

How do you classify polynomials by degree?

Back

Polynomials are classified by their degree, which is the highest power of the variable in the expression. For example, a polynomial of degree 3 is called a cubic polynomial.

Front

What is the degree of the polynomial 5x^7 - 3x^4 + 2?

Back

The degree of the polynomial is 7, as it is the highest power of x.

Front

What are the zeros of a polynomial?

Back

The zeros of a polynomial are the values of x for which the polynomial evaluates to zero. They are also known as the roots of the polynomial.

Front

How do you find the zeros of the polynomial f(x) = x^2 - 5x + 6?

Back

To find the zeros, set f(x) = 0. Factoring gives (x-2)(x-3) = 0, so the zeros are x = 2 and x = 3.

Front

What does it mean for a zero to have multiplicity?

Back

Multiplicity refers to the number of times a particular zero appears as a root of the polynomial. For example, if (x-2)^3 is a factor, then x = 2 has a multiplicity of 3.

Front

What is the end behavior of a polynomial?

Back

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

Front

What is a leading coefficient?

Back

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It influences the end behavior of the polynomial.

Front

How does the sign of the leading coefficient affect the graph of a polynomial?

Back

If the leading coefficient is positive, the graph rises to the right; if negative, it falls to the right.

Front

What is the difference between even and odd degree polynomials?

Back

Even degree polynomials have the same end behavior on both sides (both ends up or both ends down), while odd degree polynomials have opposite end behavior (one end up and the other down).