Answer:
[tex]P(x) = -0.4(x^3 - 9x^2 + 15x + 25)[/tex]
Step-by-step explanation:
Zeros of a function:
Given a polynomial f(x), this polynomial has roots [tex]x_{1}, x_{2}, x_{n}[/tex] such that it can be written as: [tex]a(x - x_{1})*(x - x_{2})*...*(x-x_n)[/tex], in which a is the leading coefficient.
Root of multiplicity 2 at x = 5 and a root of multiplicity 1 at x = -1.
This means that [tex]x_1 = x_2 = 5, x_3 = -1[/tex]
So
[tex]P(x) = a(x - x_{1})*(x - x_{2})*(x-x_3)[/tex]
[tex]P(x) = a(x - 5)*(x - 5)*(x-(-1))[/tex]
[tex]P(x) = a(x-5)^2(x+1)[/tex]
[tex]P(x) = a(x^2 - 10x + 25)(x+1)[/tex]
[tex]P(x) = a(x^3 - 9x^2 + 15x + 25)[/tex]
The y-intercept is y = -10.
This means that when [tex]x = 0, y = -10[/tex]. We use this to find a.
[tex]P(x) = a(x^3 - 9x^2 + 15x + 25)[/tex]
[tex]-10 = 25a[/tex]
[tex]a = -\frac{10}{25}[/tex]
[tex]a = -0.4[/tex]. So
[tex]P(x) = -0.4(x^3 - 9x^2 + 15x + 25)[/tex]
