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Find the roots for the given polynomial:
x5 - 3x4 - x3 + 11x2 - 12x + 4 = 0
Solution:
To solve for the roots of a given polynomial, we have to see the last term which is 4. Let's assign the factors of 4 which are 1, -1, 2, -2, 4, and -4.
First, let's check the number of positive and negative roots of a given polynomial using Descartes' Rule of Sign. Without changing the sign of x, the number of positive roots are 4 as shown below:
Next change x into -x and substitute to the given equation:
(-x)5 - 3(-x)4 - (-x)3 + 11(-x)2 - 12(-x) + 4 = 0
-x5 - 3x4 + x3 + 11x2 + 12x + 4 = 0
Since the highest degree of a given polynomial is 5, the total number of roots must be 5. There are 4 positive roots and 1 negative root of a given polynomial by Descartes' Rule of Sign. Let's see if the number of roots are correct by getting the factors using Synthetic Division, we have
x5 - 3x4 - x3 + 11x2 - 12x + 4 = 0
If you will perform the Synthetic Division method, you have to arrange the polynomial in descending order first and then consider only their coefficient. In case that any of their middle term is missing, then you have to consider 0 as the coefficient of the missing term.
Continue to do the Synthetic Division as much as you can until you have one term left on the left side. You have to do this in trial and error using the factors of the last term so that the remainder is zero at the right side.
Therefore, the roots are 1, 1, 1, 2, and -2.