Graphs of Polynomials

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 n is a non-negative integer.

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. It determines the polynomial's end behavior and the number of roots.

Front

What are the end behaviors of polynomial functions?

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 degree and leading coefficient of the polynomial.

Front

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

Back

The leading coefficient determines the direction of the graph's ends. If it is positive, the graph rises to the right; if negative, it falls to the right.

Front

What is a root of a polynomial function?

Back

A root of a polynomial function is a value of x for which f(x) = 0. Roots can be real or complex and correspond to the x-intercepts of the graph.

Front

What is the significance of the x-intercepts in polynomial graphs?

Back

The x-intercepts of a polynomial graph represent the roots of the polynomial. They indicate where the graph crosses or touches the x-axis.

Front

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

Back

A polynomial of degree n can have up to n roots, counting multiplicities. This means it can intersect the x-axis at most n times.

Front

What is the difference between even and odd degree polynomials?

Back

Even degree polynomials have symmetrical end behavior (both ends go in the same direction), while odd degree polynomials have opposite end behavior (one end goes up, the other goes down).

Front

How do you determine the number of turning points in a polynomial graph?

Back

The maximum number of turning points in a polynomial graph is one less than the degree of the polynomial.

Front

What is the factored form of a polynomial?

Back

The factored form of a polynomial expresses it as a product of its linear factors. For example, f(x) = (x - r1)(x - r2)...(x - rn), where r1, r2, ..., rn are the roots.