Find all the zeros of the function. g(x) = x5 − 5x4 + 16x3 − 28x2 + 23x − 7

Question

I am looking for help, this is the only question from my homework that I am having trouble with at the moment, I answered what I know and it has been wrong, don't know what I am doing wrong sadly.

Answer 1

The possible rational zeroes are either 1, -1, 7, -7 (rational root theorem)

g(1) = 0 meaning that 1 is 0.

Use synthetic division using 1 as the divisor. You get x^4 - 4x^3 + 12x^2 -16x+7

Using 1 as input again, we get 0.

Use synthetic division using 1 as the divisor and the quartic equation. You get x^3 - 3x^2 + 9x - 7.

Using 1 as input again, we get 0.

Use synthetic division using 1 as the divisor and the cubic equation. You get x^2 - 2x + 7.

If you want all zeros, use quadratic formula to solve for the quadratic.

the zeros are 1 (multiplicity of 3), 1+isqrt(6), 1 - isqrt(6)

Answer 2

(x - 1)³(x² - 2x + 7)