Polynomial Graphs

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 a_n ≠ 0.

Front

What does the degree of a polynomial indicate?

Back

The degree of a polynomial indicates the highest power of the variable in the polynomial. It determines the polynomial's end behavior and the maximum number of x-intercepts.

Front

What is the leading coefficient of a polynomial?

Back

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It influences the direction of the graph as x approaches positive or negative infinity.

Front

How do you find the x-intercepts of a polynomial function?

Back

To find the x-intercepts, set the polynomial equal to zero and solve for x. The solutions are the x-intercepts, where the graph crosses the x-axis.

Front

What is the significance of the zeros of a polynomial?

Back

The zeros of a polynomial are the values of x that make the polynomial equal to zero. They represent the x-intercepts of the graph and are crucial for graphing 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.

Front

What is the end behavior of a polynomial function?

Back

The end behavior describes how the values of the polynomial function behave as x approaches positive or negative infinity. It is determined by the degree and leading coefficient.

Front

How do you determine the multiplicity of a root?

Back

The multiplicity of a root refers to the number of times a particular root appears in the factored form of the polynomial. It affects the shape of the graph at the x-intercept.

Front

What is the relationship between the degree of a polynomial and the number of x-intercepts?

Back

The degree of a polynomial indicates the maximum number of x-intercepts it can have. A polynomial of degree n can have up to n real x-intercepts.

Front

What is the difference between even and odd degree polynomials?

Back

Even degree polynomials have symmetric end behavior (both ends go up or down), while odd degree polynomials have opposite end behavior (one end goes up and the other goes down).