Question

The polynomial of degree 3, P(x), has a root of multiplicity 2 at x=4 and a root of multiplicity 1 at x=−2. The y-intercept is y=−9.6
Find a formula for P(x)

Answer 1

Multiplicity is hoa many times it repeats
example
if you have roots, r1 multiplicity 2 and r2 multiliicty 3 the factors are
f(x)=(x-r1)^2(x-r2)^3


so

multiplicty 2 at x=4
multilicty 1 at x=-2
(x-4)^2(x-(-2))^1=
(x-4)^2(x+2))
expand
x^3-6x^2+32
remember
for roots, r1 and r2
(x-r1)(x-r2)
the roots don't change if you multiply whold thing by a unless a=0

so
the yint is where x=0
so the yint now is 32
32 times what=-9.6?
divide both sides by 32
what=-0.3

the polynomial is
p(x)=-0.3(x-4)²(x+2)
expanded is
p(x)=-0.3x³+1.8x²-9.6