Final Answer:
Dividing f(x) = 27x^5 - 33x^4 - 21x^3 by g(x) = 3x^2 results in 9x^3 - 11x^2 - 7x.
Step-by-step explanation:
To find f(x) / g(x), we perform polynomial division. Here's the step-by-step breakdown:
Rewrite f(x) in descending order of powers: 27x^5 - 33x^4 - 21x^3.
Divide the leading term of f(x) by the leading term of g(x): 27x^5 / 3x^2 = 9x^3.
Multiply the resulting term by g(x): 9x^3 * 3x^2 = 27x^5.
Subtract this product from f(x): (27x^5 - 33x^4 - 21x^3) - 27x^5 = -33x^4 - 21x^3.
Repeat steps 2-4 with the remaining polynomial and g(x):
Divide: -33x^4 / 3x^2 = -11x^2.
Multiply by g(x): -11x^2 * 3x^2 = -33x^4.
Subtract the product: (-33x^4 - 21x^3) - (-33x^4) = -21x^3.
Repeat steps 2-4 one last time:
Divide: -21x^3 / 3x^2 = -7x.
Multiply by g(x): -7x * 3x^2 = -21x^3.
Subtract the product: (-21x^3) - (-21x^3) = 0.
Therefore, f(x) / g(x) = 9x^3 (-11x^2) (-7x) = 9x^3 - 11x^2 - 7x.