Polynomial and Rational Functions Review

Front

What are the zeros of the polynomial function f(x) = x^3(x + 1)^2(x - 4)?

Back

The zeros are 0 (multiplicity 3), -1 (multiplicity 2), and 4 (multiplicity 1).

Front

How do you determine where a polynomial function crosses the x-axis?

Back

Set the polynomial equal to zero and solve for x. The solutions are the x-intercepts.

Front

What does it mean if a polynomial has a zero with multiplicity greater than 1?

Back

It means the graph touches the x-axis at that point but does not cross it.

Front

What is a hole in the graph of a rational function?

Back

A hole occurs at a value of x where the function is undefined due to a common factor in the numerator and denominator.

Front

If (x + 3) divides a polynomial function evenly, what can be said about the polynomial?

Back

x = -3 is a zero of the polynomial.

Front

What is the result of dividing (2x^3 + 5x^2 + 9) by (x + 3)?

Back

The result is 2x^2 - x + 3 with a remainder.

Front

What is the significance of the degree of a polynomial?

Back

The degree indicates the highest power of x and determines the polynomial's end behavior.

Front

How do you find the multiplicity of a zero?

Back

The multiplicity is the exponent of the factor corresponding to that zero in the polynomial's factored form.

Front

What is the difference between a polynomial function and a rational function?

Back

A polynomial function is a sum of terms with non-negative integer exponents, while a rational function is a ratio of two polynomial functions.

Front

What is the Remainder Theorem?

Back

The Remainder Theorem states that the remainder of the division of a polynomial f(x) by (x - c) is f(c).