Question

The polynomial of degree 3 , P ( x ) , has a root of multiplicity 2 at x = 3 and a root of multiplicity 1 at x = − 2 . The y -intercept is y = − 1.8 .

Answer 1

`\displaystyle P(x)=-(1)/(10)(x-3)^2(x+2)`

Explanation:

Multiplicity is basically how much of a factor of a polynomial appears. In this case, if we disregard the y-intercept, we can show a multiplicity of 2 at x = 3 and a multiplicity of 1 at x=-2 with
`y=(x-3)^2(x+2)`.

Of course, there will need to be a scale factor for the function so that the y-intercept will be at y = -1.8, so we can solve the following:

`-1.8=a(0-3)^2(0+2)\\-1.8=a(3)^2(2)\\-1.8=a(9)(2)\\-1.8=18a\\-0.1=a`

Thus, we can write the function as
`\displaystyle P(x)=-(1)/(10)(x-3)^2(x+2)`. See below for a visual.