what is z cube 3 plus z squared minus 7z-7 divide z squared minus 3z minus 4

To divide the polynomial (z^3 + z^2 - 7z - 7) by (z^2 - 3z - 4), we can use polynomial long division.

Step 1: Write the dividend and divisor in descending order of powers.
(z^3 + z^2 - 7z - 7) ÷ (z^2 - 3z - 4)

Step 2: Divide the highest term of the dividend by the highest term of the divisor.
Divide (z^3 / z^2) = z

Step 3: Multiply the divisor by the quotient and subtract it from the dividend.
Multiply (z^2 - 3z - 4) by z and subtract from (z^3 + z^2 - 7z - 7).

z * (z^2 - 3z - 4) = z^3 - 3z^2 - 4z
Subtract: (z^3 + z^2 - 7z - 7) - (z^3 - 3z^2 - 4z) = 4z^2 - 3z - 7

Step 4: Repeat steps 2 and 3 with the new polynomial obtained:
(4z^2 - 3z - 7) ÷ (z^2 - 3z - 4)

Divide (4z^2 / z^2) = 4

Multiply (z^2 - 3z - 4) by 4 and subtract from (4z^2 - 3z - 7).

4 * (z^2 - 3z - 4) = 4z^2 - 12z - 16
Subtract: (4z^2 - 3z - 7) - (4z^2 - 12z - 16) = 9z - 9

Step 5: The remainder obtained is 9z - 9, which has a degree of 1 (z^1).

Therefore, the quotient is z + 4, and the remainder is (9z - 9) / (z^2 - 3z - 4).

In summary, the division of (z^3 + z^2 - 7z - 7) by (z^2 - 3z - 4) is:

Quotient: z + 4
Remainder: (9z - 9) / (z^2 - 3z - 4)