The cubic polynomial f(x) is such that the coefficient of x^3 is -1. and the roots of the equation f(x) = 0 are 1, 2 and k. Given that f(x)has a remainder of 8 when divided by (x-3), find the value of k.

okay, this is what i did:

-x^3 + bx^2 + cx + d = (x-1)(x-2)(x-k)
f(3)=8
-x^3 + bx^2 +cx +d = (x-1)(x^2-kx-2x+2k)
and im stuck...

is my interpretation of the question correct? when the questions states that the coefficient of x^3 is -1, it means the polynomial is something like -x^3 + ax^2 + bx + c right???

let f(x)= -(x-1)(x-2)(x-k)

given that f(3)=8,
-(2)(1)(3-k)=8
-2(3-k)=8
-6+2k=8
2k=14
k=7

hope this helped

There's a bit of calculation mistakes there, but thanks, i know how to do it now!!!! XD

don't expand it.

let f(x) = -(x-1)(x-2)(x-k)
it should be clear that if we would expand the right side, the first term would be -x^3

given f(3) = 8 , so ...
-(2)(1)(3-k) = 9
-2(3-k) = 8
-6 + 3k = 8
3k = 15
k = 5

You're on the right track with your interpretation of the question! When the question states that the coefficient of x^3 is -1, it means that the polynomial can be written in the form of -x^3 + ax^2 + bx + c. So your understanding is correct!

Now, let's continue with your calculations. You correctly expanded (x-1)(x-2)(x-k) to -x^3 + bx^2 + cx + d, which is a good start.

To find the value of k, we can substitute x=3 into the polynomial and set it equal to 8, since we know that f(3) = 8:

-3^3 + b(3)^2 + c(3) + d = 8

Simplifying this equation, we get:

-27 + 9b + 3c + d = 8

Now, let's substitute the expanded form -x^3 + bx^2 + cx + d:

-27 + 9b + 3c + d = -3^3 + b(3)^2 + c(3) + d

Simplifying further:

-27 + 9b + 3c + d = -27 + 9b + 3c + d

This equation is true for any values of b, c, and d, which means that no matter what the values are, k can be any number. Thus, k can be any real number!

I hope this helps! Let me know if you have any more questions.

Yes, your interpretation of the question is correct. When the question states that the coefficient of x^3 is -1, it means that the cubic polynomial can be written in the form -x^3 + ax^2 + bx + c, where a, b, and c are constants.

Now, let's use the given information step-by-step to find the value of k.

Step 1: Write the cubic polynomial in factored form using the given roots:
f(x) = (x-1)(x-2)(x-k)

Step 2: Expand the factored form to get the polynomial expression:
f(x) = (x-1)(x-2)(x-k) = (x^2 - 3x + 2)(x-k) = x^3 - (3+k)x^2 + (2k+3)x - 2k

Step 3: Use the given information that f(x) has a remainder of 8 when divided by (x-3). This means that if we substitute x=3 into the polynomial, the result will be 8:
f(3) = 8

Step 4: Substitute x=3 into the polynomial expression obtained in Step 2 and set it equal to 8:
3^3 - (3+k)(3)^2 + (2k+3)(3) - 2k = 8

Simplify the equation and solve for k:
27 - (9+3k) + (6k+9) - 2k = 8
27 - 9 - 3k + 6k + 9 - 2k = 8
-3k + 6k - 2k = 8 - 27 + 9 - 9
k = -21

Therefore, the value of k is -21.