Polynomial Functions

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: $$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ where $a_n$ is not zero.

Front

What does it mean for a number to be a zero of a polynomial function?

Back

A zero of a polynomial function is a value of x that makes the function equal to zero. In other words, if $f(c) = 0$, then c is a zero of the function.

Front

How do you determine the multiplicity of a zero?

Back

The multiplicity of a zero refers to the number of times a particular zero appears as a root of the polynomial. If a polynomial can be factored as $(x - r)^m$, then r is a zero with multiplicity m.

Front

What is the degree of a polynomial?

Back

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial $f(x) = 2x^3 + 3x^2 + 1$, the degree is 3.

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. It is determined by the leading term of the polynomial.

Front

What is the factor theorem?

Back

The factor theorem states that if $f(c) = 0$, then $(x - c)$ is a factor of the polynomial $f(x)$.

Front

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

Back

To find the x-intercepts of a polynomial function, set the function equal to zero and solve for x. The solutions are the x-intercepts.

Front

What is synthetic division?

Back

Synthetic division is a simplified method of dividing a polynomial by a linear factor of the form $(x - c)$. It is faster than long division and is used to find zeros and factors.

Front

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

Back

The maximum number of x-intercepts of a polynomial function is equal to its degree. However, some x-intercepts may have multiplicities greater than one.

Front

What does it mean for a polynomial to have a repeated root?

Back

A repeated root occurs when a zero of the polynomial has a multiplicity greater than one. This means the graph touches the x-axis at that point but does not cross it.