Divide the polynomials 4x^4+4x-10

Question

by 2x^2-3
to determine the quotient and remainder.

A
The quotient is 2x2−32x^2-32x
2
−3. The remainder is 4x−194x-194x−19.

B
The quotient is 2x2+32x^2+32x
2
+3. The remainder is 4x−14x-14x−1.

C
The quotient is 2x2−32x^2-32x
2
−3. The remainder is 4x+14x+14x+1.

D
The quotient is 2x2+32x^2+32x
2
+3. The remainder is 4x−194x-194x−19.

Answer 1

To divide the polynomial \( 4x^4 + 4x - 10 \) by \( 2x^2 - 3 \), we can use polynomial long division. Let's go through the steps:

1. Divide the first term: \[ \frac{4x^4}{2x^2} = 2x^2 \]

2. Multiply: \[ 2x^2 \cdot (2x^2 - 3) = 4x^4 - 6x^2 \]

3. Subtract: \[ (4x^4 + 4x - 10) - (4x^4 - 6x^2) = 6x^2 + 4x - 10 \]

4. Repeat the process with the new polynomial \(6x^2 + 4x - 10\):

   1. Divide the first term: \[ \frac{6x^2}{2x^2} = 3 \]
   2. Multiply: \[ 3 \cdot (2x^2 - 3) = 6x^2 - 9 \]
   3. Subtract: \[ (6x^2 + 4x - 10) - (6x^2 - 9) = 4x - 1 \]

Now we have completed the division.

The quotient is \( 2x^2 + 3 \) and the remainder is \( 4x - 1 \).

Thus, the correct answer is:

B: The quotient is \( 2x^2 + 3 \). The remainder is \( 4x - 1 \).