# the function f(x) = ax^3 - 2x^2 + bx + 24 has three factors. Two of the factors are x-2 and x+3. Determine the value of a and b, and then determine the other factor

## alternate approach , assuming there is no typo.

f(x) = ax^3 - 2x^2 + bx + 24
x-2 is a factor, so
f(2) = 8a - 8 + 2b + 24 = 0
4a + b = -8

x+3 is a factor, so
f(-3) = -27a - 18 - 3b + 24 = 0
9a + b = 2

subtract them
5a = 10
a = 2 , then b = -16

f(x) = 2x^3 - 2x^2 - 16x + 24

= (x-2)(x+3)(2x - 4) , the third factor would have to be (2x - 4)

SMH

## To determine the values of a and b, we can use the given factors to find the third factor.

1. Start by writing the given factors in the form (x - r) where r is the given root.
- Factor 1: x - 2
- Factor 2: x + 3

2. Multiply the two factors together to find the third factor:
(x - 2)(x + 3) = x^2 + x - 6

At this point, we know that the third factor is x^2 + x - 6.

3. Now, we can find the value of a and b by comparing the original function with the factored form.
f(x) = ax^3 - 2x^2 + bx + 24 (Original function)
f(x) = (x - 2)(x + 3)(x^2 + x - 6) (Factored form)

Comparing the coefficients of the corresponding terms, we can equate them:

For the x^3 term:
a = 1

For the x^2 term:
-2 = 1 * (coeff of x^2 in the third factor)
-2 = 1 * 1
-2 = -2

So, b = -2.

Now, we have the values of a and b:
a = 1
b = -2

To determine the other factor, we can divide the original function by the given factors:

f(x) / (x - 2) = a(x^2 + x - 6) = (x + 3)
Dividing f(x) by (x - 2), we set x = 2 in the other factor (x + 3):
(2 + 3) = 5

Therefore, the other factor is x + 5.

## (x-2)(x+3)(x-c) = x^3 + (1-c)x^2 - (c+6)x + 6c

So that gives you
a = 1
1-c = -2
b = -(c+6)
6c = 24
solve that and you have
a = 1
b = -10
c = 4
check: (x-2)(x+3)(x-4) = x^3 - 3x^2 - 10x + 24
It appears you have a typo in your polynomial.