Question

Divide the polynomials 4x^4+4x-10 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 will perform polynomial long division.

### Step 1: Set Up the Division
We are dividing \( 4x^4 + 4x - 10 \) by \( 2x^2 - 3 \).

### Step 2: Divide the Leading Terms
Divide the leading term \( 4x^4 \) by \( 2x^2 \):
\[
\frac{4x^4}{2x^2} = 2x^2
\]
So, the first term of the quotient is \( 2x^2 \).

### Step 3: Multiply and Subtract
Now, multiply \( 2x^2 \) by \( 2x^2 - 3 \):
\[
2x^2 \cdot (2x^2 - 3) = 4x^4 - 6x^2
\]
Next, subtract this from the original polynomial:
\[
(4x^4 + 4x - 10) - (4x^4 - 6x^2) = 6x^2 + 4x - 10
\]

### Step 4: Repeat the Process
Now we divide the new leading term \( 6x^2 \) by \( 2x^2 \):
\[
\frac{6x^2}{2x^2} = 3
\]
So, the next term in the quotient is \( +3 \).

### Step 5: Multiply and Subtract Again
Multiply \( 3 \) by \( 2x^2 - 3 \):
\[
3 \cdot (2x^2 - 3) = 6x^2 - 9
\]
Now, we subtract this from our current polynomial:
\[
(6x^2 + 4x - 10) - (6x^2 - 9) = 4x - 1
\]

### Step 6: Write Down the Results
At this point, we have completed the long division since the degree of the remainder \( 4x - 1 \) is less than the degree of the divisor \( 2x^2 - 3 \).

### Final Result
The quotient is:
\[
2x^2 + 3
\]
and the remainder is:
\[
4x - 1
\]

### Conclusion
The correct option is:
**B**: The quotient is \( 2x^2 + 3 \) and the remainder is \( 4x - 1 \).