Find all rational zeros of the polynomial?

Question

P(x)=x^4 + 4 x^3 - 4 x^2 + 4 x - 5

ts rational zeros are
x_1=
x_2=
x_3=
x_4=

If the polynomial has only three rational zeros, input them at x_1, x_2 and x_3; If the polynomial has only two rational zeros, input them at x_1 and x_2; if the polynomial has only one rational zero, input it at x_1.

Answer 1

By inspection, we see that 1 is a zero of P(x).
So x-1 is a factor.
By long division or synthetic division,
the quotient is x³+5x²+x+5.
Now we set this factor to zero and solve.
x³+5x²+x+5 =0
x²(x+5) + (x+5) = 0.
(x+5)(x²+1) = 0
So the only other rational root is x = -5.
So x_1 = -5, x_2 = 1.

Answer 2

The rational zero is one. Try it. 1^4 =1 + 4 -4 + 4 -5 = 0