rational root theorem

Question

Use the rational root theorem to identify the possible roots. Verify with synthetic division and write as a product of its factors. Use the factors to find the remaining roots. Use additional paper for your work.

g(x)=x^4-6x^3+9x^2-24x+20

Answer 1

The rational root theorem is a "short cut" to help you find roots of complex polynomials without advanced technology.

First identify the leading coefficient (the coefficient of the highest degree term) and call is "q". Then identify the value of the constant (the number not containing a variable) and call it "p." The p and q letters are just arbitrary.

For your problem: x⁴-6x³+9x²-24x+20, q = 1 and p = 20

Next, find all the factors for each p and q:

factors of q = ±1

factors of p = ±1, ±2, ±4, ±5, ±10, ±20

We use ± because we are accounting for all cases to get the product (think: -1 times -1 would still equal our q value of 1).

Now we make a list (sometimes it's LONG!) of all possibilities of p/q. We are lucky this time because our q list is small, and we are really just dividing the p list by 1. This list contains all the possible rational roots of the original problem:

p/q = ±1, ±2, ±4, ±5, ±10, ±20

Now comes the tedious part. We need to check from our list of p/q which root will work for our problem. Your teacher has asked you to use synthetic division to do this, so I am assuming you know how to do that. I will demonstrate using the first root, 1:

1| 1 -6 9 -24 20

1 -5 4 -20

1 -5 4 -20 0 <----- the fact that there is a zero in the last spot means we have found a root!!

Now we can say x⁴-6x³+9x²-24x+20 = (x - 1)(x³ - 5x² + 4x - 20). The good news is, we can continue to use our list of p/q on the second listed factor here. OR you can recognize that the second factor can be factored further using grouping: x³ - 5x² + 4x - 20 = (x² + 4)(x - 5)

Now we can represent g(x) as g(x) = (x - 1)(x² + 4)(x - 5).

g(x) as two real roots x = 1 and x = 5 and two imaginary roots x = -2i and x = 2i.