Question

Find all the complex zeros for the polynomial function. show work
f(x)=x^3-7x^2+x-7

Answer 1

±i

Explanation:

Observing that the first two coefficients are the same as the last two, we can factor this function by grouping.

f(x) = (x^3 -7x^2) +(x -7) = x^2(x -7) +1(x -7)

f(x) = (x^2 +1)(x -7)

The factor x-7 has a real zero at x=7, so the complex zeros come from the quadratic factor (x^2 +1).

Setting that to zero and solving for x, we find ...

x^2 +1 = 0

x^2 = -1

x = ±√(-1) = ±i

The complex zeros are x = +i and x = -i.