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 = − 1 . The y -intercept is y = − 3.2 . Find a formula for P ( x ) .

Answer 1

The polynomial is;

P(x) = 0.2(x^3-9x^2+ 24x - 16

Explanation:

Here, we want to write the formula for the polynomial

We start by equating the roots to zero

x = 4 becomes x-4

x = -1 becomes x + 1

The expression here will be;

P(x) = a (x-4)^2(x+ 1)

P(x) = a { (x-1)(x^2-8x + 16)}

P(x) = a{ x^3-8x^2+16x-x^2+8x-16}

P(x) = a {x^3-9x^2+24x-16}

To get the value of a, set x = 0 and P(x) to -3.2

This is because the value of x at the y-intercept is 0

So we have that;

-3.2 = a(-16)

a = -3.2/-16

a = 0.2

So the polynomial will project as planned.

P(x) = 0.2(x^3-9x^2+ 24x - 16