Final answer:
To find the value of k in the polynomial f(x) = 3x³ + 3x + k, divide it by x + 3x + 3 and find the remainder. Setting the remainder equal to -74, we solve the equation to get k = -47 - 21x.
Step-by-step explanation:
To find the value of k, we need to divide the polynomial f(x) = 3x³ + 3x + k by x + 3x + 3 and find the remainder. The remainder is given as -74. Let's use polynomial long division to find k:
Dividend: 3x³ + 3x + k
Divisor: x + 3x + 3
First, divide the first term of the dividend by the first term of the divisor which gives us 3x². Multiply x + 3x + 3 by 3x² and subtract the result from the dividend:
Dividend: 3x³ + 3x + k - (3x²(x + 3x + 3)) = 3x³ + 3x + k - (3x³ + 9x² + 9x) = -9x² - 6x + k
Now, divide -9x² by x + 3x + 3 which gives us -9x. Multiply x + 3x + 3 by -9x and subtract the result from the previous step:
-9x² - 6x + k - (-9x(x + 3x + 3)) = -9x² - 6x + k - (-9x² - 27x - 27) = 21x - 27 + k
Since this is the remainder and it is given as -74, we can set it equal to -74:
21x - 27 + k = -74
Now, solve for k:
21x - 27 = -74 - k
21x = -74 - k + 27
21x = -47 - k
The equation 21x = -47 - k means that the value of k is equal to -47 - 21x.