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
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)
ah, yes - My bad. I assumed a was 1.
thank you so much guys!!!
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.