Glossary for Lesson 20: Factor Theorem

Higher-Order Polynomial Function

A polynomial function is any function that is defined by a polynomial expression.  The term higher-order polynomial generally refers to a polynomial of degree 3 or higher.  Examples of higher-order polynomial functions include the following: y=x^7 y=-x^3+4x^2+x-8 y=3x^6-8x^2+5 The graphs of polynomial functions are continuous and smooth. 

Quadratic Function

A quadratic function is a polynomial function of degree 2, also known as a second degree polynomial.  It can be written in the form f(x)=ax^2+bx+c , where a , b , and c are constants.  The simplest quadratic function is f(x)=x^2 .

Left- and Right-Hand Behavior

The left-hand behavior of a function tells us what happens to the value of the function as x tends toward -\inf , way to the left side of the graph. The right-hand behavior of a function tells us what happens to the value of the function as x tends toward \inf , way to the right side of the graph. We can also refer to the right- and left-hand behavior of a graph as its end behavior, since we’re trying to find out what is happening to the ends of the graph. For a polynomial function, as long as it is of degree 1 or greater, the end behavior on either side will either consist of the function rising way up high (approaching \inf ) or falling way down low (approaching -\inf ).

Leading Coefficient Test

We can use the leading coefficient test to find out the left- and right-hand behavior of a polynomial function based on its degree and the sign of its leading coefficient.
*If its degree is odd and its leading coefficient is positive, then the graph will fall to the left and rise to the right.  In other words, it will come up from down low on the left and rise up high on the right.
*If its degree is odd and its leading coefficient is negative, then the graph will rise to the left and fall to the right.  In other words, it will come down from up high on the left and fall down low on the right.
*If its degree is even and its leading coefficient is positive, then the graph will rise to the left and rise to the right.  In other words, it will come down from up high on the left and rise up high on the right.
*If its degree is even and its leading coefficient is negative, then the graph will fall to the left and fall to the right.  In other words, it will come up from down low on the left and fall down low on the right. 

Factor Theorem

The Factor Theorem tells us that if we have a polynomial f(x) , then (x-k) will be one of its factors if and only if f(k)=0 .  This is because if (x-k) is a factor, then plugging in k for x will make that factor evaluate to 0, which will in turn make the value of the entire function become 0.