Final answer:
To divide the polynomial (6z³ + z² + 4z - 1) by (3z + 2), use polynomial long division. The quotient is 2z² - 3z - 1/3 and the remainder is (-9z - 1)/3z.
Step-by-step explanation:
To divide the polynomial (6z³ + z² + 4z - 1) by (3z + 2), we can use polynomial long division. First, divide the first term of the polynomial by the first term of the divisor:
6z³ ÷ 3z = 2z²
Multiply the divisor by the quotient obtained:
(2z²)(3z + 2) = 6z³ + 4z²
Subtract this product from the original polynomial:
(6z³ + z² + 4z - 1) - (6z³ + 4z²) = -3z² + 4z - 1
Repeat the process with the new polynomial (-3z² + 4z - 1) and the divisor (3z + 2). The quotient obtained is -3z.
Finally, divide the constant term of the new polynomial by the first term of the divisor:
-1 ÷ 3z = -1/3z
So the quotient is 2z² - 3z - 1/3 and the remainder is -3z - 1/3z. We can write the remainder as a fraction by combining the two terms:
-3z - 1/3z = (-9z - 1)/3z