I need help understanding this math problem

Question

A cubic function has roots of -3, 1/2, 2, and a y-intercept of 6. Determine the function. Find the value of P(2).

Answer 1

I would have changed one thing.

P(x) = ( x + 3 ) ( 2x - 1 ) ( x - 2 )

multiply the 2 • ( x - 1/2 ). It looks more like an equation you would see

Answer 2

Cubic equation has roots: -3, 1/2, 2

y-intercept of 6

Determine the function and find P(2)

SOLUTION:

P(x) = a(x + 3)(x - 1/2)(x - 2)

To find "a" use the given point, (x,y) = (0,6)

Note: P(0) = 6

6 = a(0 + 3)(0 - 1/2)(0 - 2)

6 = a(3)(-1/2)(-2)

6 = a(3)

2 = a

Now, we have the whole equation:

P(x) = 2(x + 3)(x - 1/2)(x - 2)

Evaluate the function for x = 2

P(2) = 2(2 + 3)(2 - 1/2)(2 - 2)

P(2) = 2(5)(3/2)(0) = 0

This is not surprising since x = 2 is a root

Note: Roots are zero's of the function!

Also note:

Since we have the full equation,

we could evaluate it at any value of x.

P(x) = 2(x + 3)(x - 1/2)(x - 2)