Question

Divide the polynomials 4x4+4x−10
by 2x2−3
to determine the quotient and remainder.(1 point)
Responses

The quotient is 2x2−3
, the remainder is 8x+4
The quotient is 2 x squared minus 3, the remainder is 8 x plus 4

The quotient is 2x2+3
, the remainder is 4x−19
The quotient is 2 x squared plus 3, the remainder is 4 x minus 19

The quotient is 2x2+3
, the remainder is 4x−1
The quotient is 2 x squared plus 3, the remainder is 4 x minus 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. **Set up the division**:
\[
\frac{4x^4 + 0x^3 + 0x^2 + 4x - 10}{2x^2 - 3}
\]

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

3. **Multiply and subtract**:
Multiply \(2x^2\) by the entire divisor \(2x^2 - 3\):
\[
2x^2(2x^2 - 3) = 4x^4 - 6x^2
\]
Now subtract this from the original polynomial:
\[
(4x^4 + 0x^3 + 0x^2 + 4x - 10) - (4x^4 - 6x^2) = 0x^4 + 0x^3 + 6x^2 + 4x - 10
\]
This simplifies to:
\[
6x^2 + 4x - 10
\]

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

5. **Multiply and subtract**:
Multiply \(3\) by the entire divisor \(2x^2 - 3\):
\[
3(2x^2 - 3) = 6x^2 - 9
\]
Now subtract:
\[
(6x^2 + 4x - 10) - (6x^2 - 9) = 0x^2 + 4x - 1
\]
This simplifies to:
\[
4x - 1
\]

6. **Conclusion**:
Since the degree of \(4x - 1\) (which is 1) is less than the degree of the divisor \(2x^2 - 3\) (which is 2), we stop here.

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

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