Polynomial division is division with polynomials. A polynomial is the sum of a collection of terms on the form , where (real numbers) and (natural numbers). This technique is very similar to the one you learned for regular division in elementary school.
Polynomial division often appears when you are factorizing polynomials of higher degrees, such as
or when you’re solving higher degree polynomial equations.
One of the most important tricks is to guess one of the roots of the expression. That may sound a little crazy, but you’re not just pulling arbitrary numbers out of a hat, you’re making a qualified guess. More specifically, guessing a solution means trying to insert numbers like
into the expression and seeing whether it becomes .
It sounds odd to just guess at solutions, but don’t worry, because it’s pretty simple.
Theory
The division has no remainder if . This also means that if , is a solution to the equation .
The division is not without a remainder when . In this case, is not a solution to the equation , and you will get a remainder when you perform the division.
Polynomial division is easier to learn through examples than through cumbersome recipes. Here’s an example to check whether is a root, and some examples of polynomial division with explanations.
Note! Dividend is the same as numerator, and divisor is the same as denominator.
Example 1
Check if is divisible by .
You know that is divisible by if . That means you can just insert into to check.
That means you can conclude that
is not divisible by .
Example 2
Follow the steps below and make sure you understand the procedure. Try doing it yourself as well, and see whether you get the same answer!
Example 3
Follow the steps below and make sure you understand the procedure. Try doing it yourself as well, and see whether you get the same answer!
Example 4
If you have the polynomial division
what values can have so that you’re left with no remainder?
In this method, you have to perform the polynomial division and set the remainder equal to zero to find .
You set the remainder equal to and solve for , which gives you . That means there is no remainder when .
If the division has no remainder, you know that
when . That means you can insert to find .
If , the division does not have a remainder.
This is considerably easier and faster than Alternative 1!