2.3 Real Zeros of Polynomial Functions

2.3 Real Zeros of Polynomial Functions

Polynomials can be factored to find their x-intercepts (also known as their zeros). However, polynomials that are raised to high degrees can be difficult to factor. This is when long division and synthetic division can be helpful!

Long Division of Polynomials

Long division is a basic math tool that is very helpful when dealing with advanced math. For example, it can be used to divide 

by x-2.

Heres how:

Its just like normal long division. Say to yourself...

1. x (from x-2) goes into x cubed how many times? Well x squared times!! 

2. Now multiply the x squared by the x-2 and subtract that from the x cubed minus 9 x squared.

3. After that is done the x cubed term should cancel out and you should be left with -7x squared. Now bring down the next term (26x)

4. Repeat! How many times does x go into  -7x squared? -7x times! And continue until you have brought down all possible terms.

If you are left with 0, then great! Whatever you have written as your quotient is your final answer. BUT, if you are left with anything other than 0, you have a remainder. So what does that mean?

Take a look...

In this case, the remainder is -1. Instead of simply writing the quotient as your final answer, you have to remember to also add the remainder divided by the dividend. This would look like:

Another important trick for long division of polynomials is to leave space for x raised to degrees that may not be present in your dividend. For example, if the dividend is x cubed minus 1 you would write:

Notice the amount of space left!! This way, when writing the quotient and doing work for long division, all terms remain lined up properly.

Synthetic Division 

Synthetic division is a shortcut for factoring really long, complicated polynomials. Here's how it works:

Lets say you want to factor f(x)=

.

First you set up your synthetic division. 

Take each coefficient and line them up in order like so. Notice that if a term is negative, the coefficient is negative when written for synthetic division. In this example, we will be trying to see if 2 is a factor, so the 2 is written off to the side. 

The first step of synthetic division is simple. You just take the first coefficient, and bring it straight down.

Now, you take the number under the line and multiply it by the number to the left of the line. So, in this case 2*2=4.

Next, just add straight down. So, since 7+4=11, write 11 under the line. You should continue this pattern of multiplying then adding until your synthetic division looks like this:

In this case, the remainder is 0. This means that 2 is indeed a factor of this polynomial. If the number in the red box were not 0, then 2 would not be a factor of 0. 

After doing synthetic division, you are left with some numbers written under a line. Are they helpful? Turns out, they are exactly what you were looking for. These are the coefficients for the newly factored function. The newest factored form of the equation is...

To fully factor this equation, simply factor repeat synthetic division for the remaining polynomial and then factor by hand until you cannot factor it any more.

At this point a common question comes to mind... How do you know which number to try to factor with? Do you really just have to pick from an infinite amount of numbers until one magically works?

Well don't worry, there's a simple way to narrow down the list. 

To make a list of possible factors simply write: 

factors of constant / factors of coefficients of term to highest degree

So, for the synthetic division done earlier in this blog, that list would be...

It may still look like a lot of options for a polynomial that will have at most 4 real solutions, but it is still a lot less than an infinite number of options.