Rational Root Theorem

Hello my name is Hardeep, Today I will be explaining how to perform the rational root theorem. We learned this Wednesday October 16. I will explain this through the example in our notes. Pictures of the examples are at the bottom of the post, I didn't know how to post cropped pictures.

 The first step is finding the a and k values.

 a is the leading coefficient of the highest degree term, which in this case is 1. K is simply the constant or the single number at the end of the polynomial, which is 6.

 The next step is to find the roots. This can be done by dividing the factors of k by the factors of a. The factors of a are just 1. The factors of k are 1,2,3,and 6.

Now we will divide 1/1, 2/1, 3/1, and 6/1. We end up with 1,2,3,6 as our roots. These are the numbers we would have plugged into our equation to solve for zero, but we don't have to do that yet, so onto example 2.

Part a) asks us to find the possible roots of 3x3-4x2-5x+2, we will do what we did previously. First we find a and k, in this instance it is a=3 and k=2.

 Next we will find the factors of a and k. The factors of a are 1 and 3. The factors of k are 1 and 2. 

From here we will divide the factors of k by the factors of a. 1/1=1, 2/1=2, 1/3=1/3, 2/3=2/3. We end up with our roots which are +-1, +-2, +-1/3, +-2/3.

 The next step is to substitute our roots into the polynomial function. P(1)= 3(1)3 - 4(1)2 - 5(1) + 2 = 3-4-5+2 = -4 This root did not work so it is not a root of this polynomial. We will continue using the next root which is -1, this step is trail and error.

 P(-1)= 3(-1)3 - 4(-1)2 - 5(-1) + 2 = -3-4+5+2=0 This one worked and so it is one of the roots of this polynomial. The polynomial must equal 0.

 Step 4 is to perform division, I will be using synthetic division.

 -1/3 -4 -5 2
        -3  7-2
-------------------
     3 -7  2 0

 We end up with 3x2 - 7x + 2, from here we will factor this and end up with (3x-1)(x-2)

 Step 6 is to combine all of the factors that we used. (x+1)(3x-1)(x-2)=0

We now find the roots by equating the factors to zero. x+1=0 becomes x=-1, 3x-1=0 becomes x=1/3 and x-2=0 becomes x=2. The roots are x=-1 x=1/3 x=2.

This is all there is to the rational root theorem. If you were keeping up you would have noticed that we completed part b) as we found the roots. The answer to b) is (x+1)(3x-1)(x-2).





Example 3 is the same as the previous example except for a few small details, I will elaborate on these but skim through the rest.

Example 3: Factor x5 + 3x4 - x - 3

Find a and k, a=1   k=-3

Find the roots, +-1, +-3

Now we plug in our roots and equate to zero.

(1)5 + 3(1)4 - 1 - 3 = 0

x4 + 4x3 + 4x2 + 4x + 3

Because our polynomial is a quintic function we must bring it down to quadratic so we will find 3 roots in total.

(-1)5 + 3(-1)4 +1 - 3 = 0

x3 + 3x2 + x + 3

(-3)5 + 3(-3)4 + 3 - 3 = 0

x2 + 1

Our 3 roots are (x-1)(x+1)(x+3)(x2 + 1)







Example 2                                                                   Example 3