Final answer:
To divide the polynomial (40y)⁴-(5y)³-(30y)²-10y by the monomial 5y, use polynomial long division. The quotient is 8y³ - y² - 6y - 2.
Step-by-step explanation:
To divide the polynomial (40y)⁴-(5y)³-(30y)²-10y by the monomial 5y, you need to use polynomial long division.
Step 1: Divide the first term of the polynomial by the monomial.
(40y)⁴ ÷ 5y = 8y³
Step 2: Multiply the monomial by the result from Step 1, and subtract it from the original polynomial.
(40y)⁴ - (8y³ * 5y) = (40y)⁴ - 40y⁴ = 0
Step 3: Repeat Steps 1 and 2 with the remaining terms of the polynomial.
(-5y)³ ÷ 5y = -y²
(-5y)³ - (-y² * 5y) = (-5y)³ + 5y³ = 0
(-30y)² ÷ 5y = -6y
(-30y)² - (-6y * 5y) = (-30y)² + 30y² = 0
(-10y) ÷ 5y = -2
(-10y) - (-2 * 5y) = (-10y) + 10y = 0
Step 4: The final quotient is the sum of the quotients obtained in each step.
Final quotient: 8y³ - y² - 6y - 2