139k views
1 vote
The polynomial degree 3, P(x), has a root of multiplicity 2 at x = 5 and a root of multiplicity 1 at x = -1. The y-intercept is y = -10.

Find a formula for P(x).

1 Answer

7 votes

Answer:


P(x) = -0.4(x^3 - 9x^2 + 15x + 25)

Explanation:

Zeros of a function:

Given a polynomial f(x), this polynomial has roots
x_(1), x_(2), x_(n) such that it can be written as:
a(x - x_(1))*(x - x_(2))*...*(x-x_n), 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
x_1 = x_2 = 5, x_3 = -1

So


P(x) = a(x - x_(1))*(x - x_(2))*(x-x_3)


P(x) = a(x - 5)*(x - 5)*(x-(-1))


P(x) = a(x-5)^2(x+1)


P(x) = a(x^2 - 10x + 25)(x+1)


P(x) = a(x^3 - 9x^2 + 15x + 25)

The y-intercept is y = -10.

This means that when
x = 0, y = -10. We use this to find a.


P(x) = a(x^3 - 9x^2 + 15x + 25)


-10 = 25a


a = -(10)/(25)


a = -0.4. So


P(x) = -0.4(x^3 - 9x^2 + 15x + 25)

User Athi
by
7.3k points