SOLUTION: Id have a question; I should find out what the possible rational roots of x^3 - 4x^2 + 2x + 1 are, I am not sure if 3.303, -0.303, 1 are right and I totally dont understand ho

Algebra ->  Square-cubic-other-roots -> SOLUTION: Id have a question; I should find out what the possible rational roots of x^3 - 4x^2 + 2x + 1 are, I am not sure if 3.303, -0.303, 1 are right and I totally dont understand ho      Log On


   

Question 205169: Id have a question; I should find out what the possible rational roots of x^3 - 4x^2 + 2x + 1 are, I am not sure if 3.303, -0.303, 1 are right and
I totally dont understand how to use synthetic substitution, to find out which of the possible rational roots are actually a root of the equation. Could someone please help me with this?
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
I should find out what the possible rational roots of x^3 - 4x^2 + 2x + 1 are, I am not sure if 3.303, -0.303, 1
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It's easy to determine that 1 is a root. Then divide by x-1
--> x^2 - 3x - 1
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-3x%2B-1+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-3%29%5E2-4%2A1%2A-1=13.

Discriminant d=13 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--3%2B-sqrt%28+13+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-3%29%2Bsqrt%28+13+%29%29%2F2%5C1+=+3.30277563773199
x%5B2%5D+=+%28-%28-3%29-sqrt%28+13+%29%29%2F2%5C1+=+-0.302775637731995

Quadratic expression 1x%5E2%2B-3x%2B-1 can be factored:
1x%5E2%2B-3x%2B-1+=+%28x-3.30277563773199%29%2A%28x--0.302775637731995%29
Again, the answer is: 3.30277563773199, -0.302775637731995. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-3%2Ax%2B-1+%29