polynomial question...?

Question

Find all rational roots of the polynomial:
P(x) = 5 x ^4 - 27 x ^3 + 40 x ^2 - 22x + 4 , then find the
irrational roots, if any.

a. 2/5, 1, and 2 ± √2
b. 2/5 and 1 ± √2
c. -1/2, 1/4 and (3 ± √5)/2
d. 1/3 and 1
e. -2/3 , 5/2 and 3 ± √3

Answer 1

Answer: A

the coefficients add up to 0.. thus 1 is a root.

5x64 - 27x^3 + 40x^2 -22x + 4
= (x-1)(5x^3-22x^2+18x-4)

This time only choices a and d are left.
Now on the other factor cannot have 1/3 as its root. Since the constant term is 4 and the coefficient of the highest term is 5.
The roots are p/q where p is a factor of 4 and q is a factor of 5.
Thus the answer is a. §

Then that means the plausible choice for rational based on the only choice left is 2/5:

Expression
= (x-1)(5x-2)(x^2 - 4x + 2)

To get the roots of (x^2 - 4x + 2):
x^2 - 4x + 2 = 0
x^2 - 4x + 4 = 2
(x - 2)^2 = ±√2
x = 2 ± √2.

Answer 2

the solution is a) as 2/5 and 1 are roots and by sythetic division you can come to 2+-sqrt2

Answer 3

the solution is a)

if you have a TI 89 calculator the function csolve( expression = 0,x) will return all real and complex roots of the expression. if you don't own a TI 89 you can get an emulator for it at:

http://mathematics.mc.maricopa.edu/seims/TI-Emulator.htm