Question 208412
Find all roots by synthetic division and using the remainder theorem and the factor theorem. 
1.) x^4-4x^3+4x-1=0
Since the coefficients add up to zero, one is a root.
Use synthetic division to find others:
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1)....1....-4.....0...4....-1
.......1....-3....-3...1...|..0

The Quotient is f(x) = x^3 - 3x^2 -3x + 1
f(-1) = -1 -3 +3 + 1 = 0
so, -1 is a root
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-1).... 1....-3....-3....1
.........1.....-4....1...|..0

Now use the quadratic formula to find
the roots of x^2 -4x + 1

x = [4 +- sqrt(16 - 4*1*!)]/2
x = [4 +- sqrt(12)]/2
x = 2 + sqrt(3) or x = 2 - sqrt(3)
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So there you have the 4 roots.
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2.) 4x^4+8x^3-7x^2-21-9=0
I graphed f(x) to find a Real Number root to start the process.
There are no "neat" roots so I'll leave this to you as it is
getting too late here.
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Cheers,
Stan H.