Question

I am familiar with this type of problem but can't seem to get the right answer.

Use the given zero to find the remaining zeros of each function

f(x)=x^4-9x^3+7x^2-91x-348,zero 5-2i

I normally would use synthetic division with the root 5-2i bringing it down to x^3. Then synthetic with 5+2i, bring it down to a quadratic solve the quadratic then I'll have zeros, but I can't get past the 5-2i with the synthetic division. Every time I do it I can't get it to equal zero. I have done it multiple times and it won't work.

Answer 1

One property of complex roots is that they always come in conjugate pairs.
So if one root is 5-2i, there will be another 5+2i
so there are two factors,
(x - 5 - 2i) and (x - 5 + 2i)

I multiplied these and go
(x^2 - 10x + 29)

Now do a long division of
(x^4-9x^3+7x^2-91x-348) by (x^2 - 10x + 29)
That should leave you with a quadratic, that can be solved for 2 more roots.
Let me know if it worked for you.