The polynomial of degree 3, P(x), has a root of multiplicity 2 at x = 2 and a root of multiplicity 1 at x = - 5 . The y-intercept is y = - 10

Question

create a polynomial equation solving for p(x)

Answer 1

P(x) = a(x - 2)²(x + 5), a ≠ 0

 

Use the given point (0,-10) to find a.

 

-10 = a(0 - 2)²(0 + 5) = a(4)(5) = 20a

a = -10/20 = -1/2

 

P(x) = (-1/2)(x - 2)²(x + 5)

 

You can expand this if you wish.

Answer 2

A root x=2 with multiplicity of 2 has a factor of (x - 2)².

A root x=-5 with multiplicity of 1 has a factor of (x + 5).

 

So far, your function is

 

P(x) = C(x - 2)²(x + 5)

 

Using the initial value condition, we can solve for C.

 

-10 = C(-2)²(5)

-10 = 20C

-1/2 = C

 

P(x) = (-1/2)(x - 2)²(x + 5)