find how many real roots of, counting multiplicities x(x^2+16)(x^2-x-6)^2=0

Question

find how many real roots of, counting multiplicities  x(x^2+16)(x^2-x-6)^2=0

please show how to get the answer. thank you

oobleck · Answer

x=0 has 1 root
x^2+16 = 0 has no real root
x^2-x-6 has 2 real roots and it is squared, so each is repeated.
So, add 'em up

The reason we always set everything to zero is that if any factors are multiplied to get zero, at least one of the factors must be zero.