If (x – 3) divides evenly into x3 – x2 – kx + k, find k. (2 marks)
To find the value of k, we need to apply the concept of polynomial division and use the fact that (x - 3) divides evenly into x^3 - x^2 - kx + k.
The process of polynomial division involves dividing the given polynomial (x^3 - x^2 - kx + k) by the divisor (x - 3) to obtain a quotient and a remainder.
Let's perform the polynomial division:
Step 1: Set up the division:
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(x - 3) | x^3 - x^2 - kx + k
Step 2: Divide the first term of the dividend by the first term of the divisor:
x^3 / (x - 3) = x^2
Step 3: Multiply the divisor by the quotient obtained in step 2:
x^2 * (x - 3) = x^3 - 3x^2
Step 4: Subtract the result obtained in step 3 from the dividend:
(x^3 - x^2 - kx + k) - (x^3 - 3x^2) = (2x^2 - kx + k)
Step 5: Bring down the next term from the dividend:
(2x^2 - kx + k) becomes (2x^2 - kx + k)
Step 6: Repeat steps 2-4 with the new dividend:
2x^2 / (x - 3) = 2x
2x * (x - 3) = 2x^2 - 6x
(2x^2 - kx + k) - (2x^2 - 6x) = (6x - kx + k)
Step 7: Continue repeating steps 2-4 until all terms have been divided:
Now we continue dividing the new dividend (6x - kx + k) by (x - 3).
Step 8: Perform the division:
(6x / (x - 3) = 6
6 * (x - 3) = 6x - 18
(6x - kx + k) - (6x - 18) = (-kx + 18 + k)
Step 9: There are no other terms left to divide. We now have the remainder, (-kx + 18 + k), and according to the problem, (x - 3) divides evenly, meaning the remainder should be zero.
Since the remainder is (-kx + 18 + k), we can set (-kx + 18 + k) equal to zero and solve for k:
-kx + 18 + k = 0
Rearranging the equation, we have:
-kx + k = -18
Factoring out k:
k(-x + 1) = -18
Dividing both sides of the equation by (-x + 1):
k = -18 / (-x + 1)
Therefore, the value of k is -18 / (-x + 1)