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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 first need to perform the division.
1. Rewrite the expression: (z^2 - 4z - 3) ÷ (z + 2z^2 + z - 12)
2. Combine like terms in the numerator: z^2 - 4z - 3
3. Combine like terms in the denominator: z + 2z^2 + z - 12 = 2z^2 + 2z - 12
4. Perform polynomial division (long division or synthetic division) to simplify the division.
(z^2 - 4z - 3) ÷ (2z^2 + 2z - 12)
Now, perform polynomial division to find the quotient:
First, divide z^2 by 2z^2, which gives 1/2
Multiply the divisor (2z^2 + 2z - 12) by 1/2 to get z + z - 6
Subtract this from the numerator:
(z^2 - 4z - 3) - (z + z - 6) = z - 3
There is a remainder of -3 left
So, the quotient is z - 3 with a remainder of -3. The restrictions on the variable z are that z cannot be equal to -2 or 2.