Finding real and imaginary roots from polynomials

Question

I have a worksheet in my hon. algebra 2 class and the question is:

State the possible number of real roots for each equation. Then find all roots.

The 1st question is:

$(x-7)(x+1)(x-1) = 0$

So would there be 3 real roots/0 imaginary roots, and the actual roots would be $7, -1,$ and $1$ since that's where it intercepts the X?

Also, another question I have, if a polynomial is squared like:

$(x-1)^2(x+7) = 0$

Would that mean the $(x-1)^2$ would be an imaginary root which would make the answer 1 real root/1 imaginary root and then $1, -7$ as the roots?

Thanks a lot!

Answer 1

If a polynomial has a factor such as $(x-a)^n$ it is named as multiplicity, not an imaginary root. Imaginary root is when delta<0.

For example let $(x^2+1)(x-2)^2=0$

Here you have imaginary roots $i$ and $-i$ from $(x^2+1)$ and double roots $2$ and $2$ from $(x-2)^2$.