Polynomial Equations

Solving a polynomial equation can be done algebraically if it is factorable or graphically if it not. Note that a polynomial function is y = P(x) and a polynomial equation is when P(x) = 0. 


If the polynomial equation is factorable, the roots are determined by solving each factor when it is equal to zero. For example, 

The x-intercepts of the graph are 0 and 4/7. The x-intercepts of a polynomial function graph correspond to the real roots of the polynomial equation P(x) = 0. Therefore, the roots for the polynomial equation in the example above is also 0 and 4/7.


P(x) can also be solved using the factor theorem. To solve the cubic x³ - x² + x -1 = 0,  
we first figure out the values that should be tested using the rational zero theorem. If P(x) is a polynomial function with integer coefficients and x = b/a is a zero of P(x), then b is a factor of the constant term while a is a factor of the leading coefficient. Furthermore, ax - b is a factor of P(x). Back to the equation, since a=1 and b=-1, the only possible values of b/a is 1. By substitution, we see that x=1 is a root thus (x-1) is a factor. We then find the other factor by dividing with (x-1) which produces the quotient x² +1. Thus, 

x³ - x² + x - 1 = (x - 1)(x² + 1) = 0

x - 1 = 0

x² + 1 = 0

Since a square root of a negative value is not a real number, we do not get a real root from x² + 1  as it is a complex root. So the only real root is x=1. If a polynomial equation has real coefficients, then either all roots are real or there are an even number of non-real complex roots, in pairs.

On the other hand, if the polynomial equation is not factorable, the roots can be determined from the graph by examining the x-intercepts. By graphing the function using a graphing calculator, you can get a sense of where the roots are and how many real roots exist when it crosses or touches the x-axis.


A set of functions with the same characteristics is a family of functions. When talking about a family of a polynomial function, we are talking about the general equation of the function. Polynomial functions that have the same x-intercepts belong to the same family. 

Remainder Theorem and Factor Theorem

We started Chapter 2 with long division. Easy, right? Well this time we learned dividing a polynomial function with a binomial. In the following example, P(x) is divided by the factor x-4.

The quotient for the example above is  x² + 4x + 9 with a remainder of 30. Thus, x³ – 7x – 6  =   (x – 4) (x² + 4x + 9)  +  30. When dividing, just remember to do so carefully to avoid mistakes. 

The Remainder Theorem is an easier way to find the remainder. Instead of using the long division method, take x=b from the factor x-b and substitute it in the polynomial function. From the example above, b=4 so we solve P(4) to find the remainder. 

P(4) = (4 – 4)((4)² + 4(4) + 9) + 30 

                                                              = (0)(16 + 16 + 9) + 30       

                                                              = 0 + 30       

                                                              = 30

Therefore, the remainder theorem:

When you divide a polynomial P(x) by x-b the remainder R will be P(b).

Now, if the remainder is 0, then x-b is a factor of the polynomial. Just as with the Remainder Theorem, the point here is not to do the long division of a given polynomial by a given factor. The same method is used as the Remainder Theorem simply by substituting b to find P(b). The Factor Theorem is just a result of the Remainder Theorem. It's purpose is to check for a zero remainder to determine the factors of the polynomial. 

And so we have: 

When P(b)=0 then x-b is a factor of the polynomial P(x).

Knowing that x-b is a factor of P(x) is useful as it tells you that b is an x-intercept of the function.

If you are still confused, the video below explains the difference between the Remainder Theorem and the Factor Theorem.

In addition to few examples of both the Remainder Theorem and Factor Theorem, this websites provides an exercise for you to work on. Yes, answers are given. I hope this post has helped you in a way. Thanks for reading. :)