Question

Divide the polynomial 12.? + 22 - 9 by 4 - 2. Then express the polynomial as the product of the divisor and the quotient plus the remainder.

Answer 1

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:
1. Divide \( 12x^2 \) by \( 2 \), which gives \( 6x^2 \).
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:
\[
12x^2 + 13 = 2(6x^2 + \frac{13}{2}) + 0
\]