How do you determine a root and perform long division on the polynomial x5-2x4-4x3 8x2 x-20?

Answer 1

Long division and determining a root are two different things in this case.Long division: First you need to know what you want to divide it by. "Long division" in this case means that you divide one polynomial by another one.

Finding a root: There is no simple formula to find roots of polynomials of degree 5 or more - and even for degree 3 and 4, it's rather complicated to find the exact value. In practice, several iterative methods are used, that can get you the roots to any desired precision. But all those methods are quite complicated; much beyond what you normally learn in high school.

Answer 2

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As a result of this fiasco, the equation which I can see is "x5-2x4-4x3 8x2 x-20".

I have tried to interpret this as one of:

(A) f(x) = x^5 - 2x^4 - 4x^3 + 8x^2 + x - 20

(B) f(x) = x^5 - 2x^4 - 4x^3 - 8x^2 + x - 20

(C) f(x) = x^5 - 2x^4 - 4x^3 + 8x^2 - x - 20

(D) f(x) = x^5 - 2x^4 - 4x^3 - 8x^2 - x - 20

that is, all the sign combinations for the terms in x^2 and x.

If any of these had an integer root then it would have to be a factor of 20 so it would have to belong to the set {-20, -10, -5, -4, -2, -1, 1, 2, 4, 5, 10, 20}. Had that been the case then if I substituted one of these values for x in the four equations, they should have given the value 0. No such luck!

So, where do you go from there? It is possible to show, graphically, that there is only one real root and that

(A) f(2) = -12 and f(4) = 368. Since the f(x) is continuous, the change of sign indicates there is a root between 2 and 4. However, since there is no factor of 20 between 2 and 4, there is no integer root. You can continue to narrow the interval down by checking for a sign change in f(x). I stopped at a root between 2.700 and 2.716

(B) f(2) = -82 and f(4) = 112. Since the f(x) is continuous, the change of sign indicates there is a root between 2 and 4. However, since there is no factor of 20 between 2 and 4, there is no integer root. You can continue to narrow the interval down by checking for a sign change in f(x). I stopped at a root between 3.729 and 3.733

(C) f(2) = -22 and f(4) = 360. Since the f(x) is continuous, the change of sign indicates there is a root between 2 and 4. However, since there is no factor of 20 between 2 and 4, there is no integer root. You can continue to narrow the interval down by checking for a sign change in f(x). I stopped at a root between 2.777 and 2.791

(D) f(2) = -86 and f(4) = 104. Since the f(x) is continuous, the change of sign indicates there is a root between 2 and 4. However, since there is no factor of 20 between 2 and 4, there is no integer root. You can continue to narrow the interval down by checking for a sign change in f(x). I stopped at a root between 3.751 and 3.755

In any of the above cases you can use numerical methods, such as the Newton-Raphson method to estimate the root. In any case, given the fact that root may not be a simple rational number, long division will be of no use.