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Question 208412This question is from textbook ALGERA
: pls. help me to solve this equation I really dont know how to get the root
Using the Descartes’ Rule of signs, give by means of tabulation, all the possible combinations of the number of positive roots, negative roots and imaginary roots.
Find all roots by synthetic division and using the remainder theorem and the factor theorem.
1.) x^4-4x^3+4x-1=0
2.) 4x^4+8x^3-7x^2-21-9=0
thank you very much for your help
This question is from textbook ALGERA
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! 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:
-----
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.
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