Question

Divide the polynomial 4x4+4x−10 by 2x2−3 to determine the quotient and remainder. (1 point) Responses The quotient is 2x2−3 and the remainder is 4x+1 The quotient is 2 x squared minus 3 and the remainder is 4x+1 The quotient is 2x2+3 and the remainder is 4x-19 The quotient is 2 x squared plus 3 and the remainder is 4x-19 The quotient is 2x2+3 and the remainder is 4x-1 The quotient is 2 x squared plus 3 and the remainder is 4x-1 The quotient is 2x2−3 and the remainder is 4x-19

Answer 1

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

1. **Set up the long division**: 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 by \(2x^2\)**:
\[
2x^2(2x^2 - 3) = 4x^4 - 6x^2
\]

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

4. **Repeat the process for \(6x^2 + 4x - 10\)**:
Divide the leading term \(6x^2\) by the leading term \(2x^2\):
\[
\frac{6x^2}{2x^2} = 3
\]

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

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

Now, we can write the result of the division:
\[
4x^4 + 4x - 10 = (2x^2 + 3)(2x^2 - 3) + (4x - 1)
\]

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

So the correct response is:
**The quotient is \(2x^2 + 3\) and the remainder is \(4x - 1\)**.