secondary (10-12))

can you help me with this problem?

Show that (x+2) is a factor of f(x) where:

f(x) = px3 + (4p+2)x2 + (5p+4)x + 2p

Find the range of values for which the equation f(x) = 0 has 3 distinct real roots



The first part of this problem is an application of the Factor Theorem.

Factor Theorem: (x - a) is a factor of the polynomial p(x) if and only if
p(a) = 0.

Hence to show that (x+2) is a factor of your polynomial f(x) you only need verify that f(-2) = 0.

For the second question I would divide f(x) by (x + 2) to get

f(x) = (x + 2) g(x)

and then look at the quadratic g(x).