To divide the polynomial \( 12x^2 + 22 - 9 \) by \( 4 - 2 \), we first simplify the divisor. The divisor \( 4 - 2 \) simplifies to \( 2 \).
Now we have: \[ \text{Dividend: } 12x^2 + 22 - 9 = 12x^2 + 13 \] \[ \text{Divisor: } 2 \]
Next, we perform the division: \[ \frac{12x^2 + 13}{2} \]
This division can be carried out as follows:
- Divide \( 12x^2 \) by \( 2 \), which gives \( 6x^2 \).
- Divide \( 13 \) by \( 2 \), which gives \( \frac{13}{2} \).
So the quotient is: \[ 6x^2 + \frac{13}{2} \]
Since we are dividing by a constant (2), there is no remainder other than the constant terms, which will be \( 0 \).
Now, expressing the polynomial as the product of the divisor and the quotient plus the remainder: \[ 12x^2 + 13 = 2 \left( 6x^2 + \frac{13}{2} \right) + 0 \]
Thus, the final answer is: \[ 12x^2 + 13 = 2(6x^2 + \frac{13}{2}) + 0 \]
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