Divide the polynomial 4x4+4x − 10 by 2x2−3

Question

to determine the quotient and remainder.(1 point)
Responses

The quotient is 2x2+3.
The remainder is 4x - 19.
The quotient is 2 x squared plus 3 pointThe remainder is 4x - 19.

The quotient is 2x2−3.
The remainder is 4x - 19.
The quotient is 2 x squared minus 3 pointThe remainder is 4x - 19.

The quotient is 2x2+3.
The remainder is 4x - 1.
The quotient is 2 x squared plus 3 pointThe remainder is 4x - 1.

The quotient is 2x2−3.
The remainder is 4x + 1.

Answer 1

To divide the polynomial \( 4x^4 + 4x - 10 \) by \( 2x^2 - 3 \), we can use polynomial long division.

1. Divide the leading term of the dividend \( 4x^4 \) by the leading term of the divisor \( 2x^2 \): \[ \frac{4x^4}{2x^2} = 2x^2 \]

2. Multiply the entire divisor \( 2x^2 - 3 \) by \( 2x^2 \): \[ 2x^2(2x^2 - 3) = 4x^4 - 6x^2 \]

3. Subtract this from the original polynomial: \[ (4x^4 + 4x - 10) - (4x^4 - 6x^2) = 4x + 6x^2 - 10 = 6x^2 + 4x - 10 \]

4. Now divide the leading term \( 6x^2 \) by the leading term of the divisor \( 2x^2 \): \[ \frac{6x^2}{2x^2} = 3 \]

5. Multiply the entire divisor \( 2x^2 - 3 \) by \( 3 \): \[ 3(2x^2 - 3) = 6x^2 - 9 \]

6. Subtract this from the current remainder: \[ (6x^2 + 4x - 10) - (6x^2 - 9) = 4x - 10 + 9 = 4x - 1 \]

Now we have completed the division. The quotient is \( 2x^2 + 3 \), and the remainder is \( 4x - 1 \).

Thus, the correct response is:

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