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What is the quotient in simplest form? State any restrictions on the variable.
z2−4z−3÷z+2z2+z−12
To find the quotient in simplest form, we need to perform polynomial division.
Step 1: Rewrite the expression as a fraction with the numerator being z^2 - 4z - 3 and the denominator being z + 2z^2 + z - 12.
Step 2: Divide the first term of the numerator by the first term of the denominator: z^2 / z = z. Write this result above the line.
Step 3: Multiply the entire denominator by z and subtract this from the numerator:
(z + 2z^2 + z - 12) * z = z^2 + 2z^2 + z^2 - 12z = 4z^2 - 12z
(z^2 - 4z - 3) - (4z^2 - 12z) = -3z + 12z = 9z
Step 4: Bring down the next term from the numerator: -3
Step 5: Repeat steps 2-4 for the new numerator:
9z / z = 9
(z + 2z^2 + z - 12) * 9 = 9z + 18z^2 + 9z - 108
(-3 - 9z) - (9z + 18z^2 + 9z - 108) = -3 - 9z - 9z - 18z^2 - 9z + 108 = 105
Step 6: Since the degree of the numerator is lower than the degree of the denominator, the quotient is complete.
Therefore, the quotient in simplest form is z + 9 with a remainder of 105. There are no restrictions on the variable.