Consider f(x) = x^3 - 2x^2 - 5x + k
Find the value of k if (x+2) is a factor of f(x)
I'm not sure if this is the answer because it says this question is worth 6 marks:
f(-2) = (-2)^3 -2(-2)^2 -5(-2) +k = 0
= -8 -8 +10 + k
= -16+10 +k
k = 6
The solution given in the question itself is definitely valid is correct.
As (x + 2) is one of the factors of f(x), the value of (x + 2) when x is equal to -2 is zero.
Therefor, when x = -2, we can equate f(-2) to 0 and then solve the equation for value of x, as has been done. However there are other ways of solving this question also. One such way is given below.
f(x) = x^3 - 2x^2 - 5x - k
and that (x + 2) is a factor of f(x)
Therefore if we divide f(x) bu (x + 2) the remainder should be 0.
Dividing f(x) by (x + 2) we get:
f(x)/((x + 2) = (x^3 - 2x^2 - 5x - k)/(x + 2)
= (x^2 - 4x + 3) + (k - 6)/(x + 2)
As k-6 in the above expression is a constant, it is the remainder. This remainder has to be 0.
k - 6 = 0
k = 6
k = 6
To find the value of k if f(x) = x^3 - 2x^2 - 5x + k is divisible by (x+2).
If x^3 - 2x^2 - 5x + k is divided by x+2, the n you are getting a n expression, Q(x) as quotient and a remainder, R. So
f(x) /(x+2) = Q(x)+ R/(x+2). Or
f(x) = (x+2) Q(x)+R. Putting x = -2, we get:
f(-2) =(-2)^3-2(-2)^2-5(-2)+k = (-2+2)*Q(-2)+R.
f(-2) = -8-8+10+k = 0+R. But R should be zero as x+2 divides f(x) without remainder. Therefore,
f(-2) =-6+k=0. Or
Note: What you have done is perfectly alright. Only that you could have added a little explanation.