# Algebra II/Polynomial Functions

A **polynomial function** is a function compromising of more than one non-negative integer powers of *x*. An example of this would be:

- 3x
^{2}+ 4x - 5 - 8x
^{4}+ 12x^{3}- 2x^{2}+ 5x - 5 - 6x
^{7}- 5

A **degree** is determined by whichever power is the highest. Example: The first polynomial, 3x^{2} + 4x - 5, has as the highest degree since it is the highest power in the polynomial. The leading coefficient is the coefficient with the highest degree (**not** value). Example: The second polynomial, 8x^{4} + 12x^{3} - 2x^{2} + 5x - 5, has "8" as its leading coefficient since it is the number that is associated with the highest degree, being "4".

A **quadratic** is a polynomial with the highest degree being . A **cubic polynomial**, or simply "cubic", is a polynomial with the highest degree being .

## Graphs

[edit | edit source]Watch closely how the degrees are related to the **zeroes** of the function (the intersection of the x-axis by the graph). They seem to correlate each other (be the exact same as each other).

Observe the **end behavior**. Which direction does x or f(x) lead to? Positive or negative infinity (take note that the functions go infinity and never stop at a certain point)? This is represented by x→+∞ and x→−∞ (x-approaching positive infinity and x-approaching negative identity, respectively). The **degree** and **leading coefficient** of a polynomial function determine the end behavior of the polynomial function graph. See Varsity Tutors - End Behavior of a Function (examples) for more examples of end behavior of functions.

- If the x-axis was intersected twice or thrice, it has two/three real zeroes.
- If the x-axis was not intersected at all, its roots are imaginary.
- If the function was tangent to the x-axis, this means it is a
**double root**, which represents two zeroes at the same number (x-axis point).

A function is **even** if both sides of the function go the same direction, while a function is **odd** if both sides of the function go in opposite directions.

You determine the number of degrees and the total number of solutions for a function by counting the turns of the graph and adding one to it.

### Maximum and Minimum Points

[edit | edit source]The **maximum point** is the highest point (in terms of the y-coordinate) on the function. The **minimum point** is the opposite. These are often referred to as "turning points". These are the points on which you count the turns.