Question

Divide the polynomials 4x
4
+4x−10 by 2x
2
−3 to determine the quotient and remainder.
The quotient is □ and the remainder is □.

Options:
A. 2x
2
+3
B. 4x−1
C. 2x
2
−3
D. 4x−19

Answer 1

Are you sure

Answer 2

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

1. **Divide the first term of the dividend by the first term of the divisor:**
\[
\frac{4x^4}{2x^2} = 2x^2
\]
This is the first term of the quotient.

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

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

4. **Now repeat the process with the new polynomial \( 6x^2 + 4x - 10 \):**
\[
\frac{6x^2}{2x^2} = 3
\]
This is the next term of the quotient.

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

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

Now, since \( 4x - 1 \) has a degree less than \( 2x^2 - 3 \), we stop here.

**The final result of the division is:**
- Quotient: \( 2x^2 + 3 \)
- Remainder: \( 4x - 1 \)

Putting it all together, we find:
\[
\text{The quotient is } 2x^2 + 3 \text{ and the remainder is } 4x - 1.
\]

However, we need to check the options provided for the values.

**Options given:**
A. \( 2x^2 + 3 \)
B. \( 4x - 1 \)
C. \( 2x^2 - 3 \)
D. \( 4x - 19 \)

Thus, the appropriate selections corresponding to the completed division are:
- The quotient is \( 2x^2 + 3 \) (Option A).
- The remainder is \( 4x - 1 \) (which is not directly listed as an option, but it corresponds to B when the representation is simplified).

Final output:
- The quotient is **2x^2 + 3** and the remainder is **4x - 1**.

Answer 3

Let's carefully go through the polynomial long division process again to ensure accuracy. We need to divide \( 4x^4 + 4x - 10 \) by \( 2x^2 - 3 \).

### Step 1: Setup the Division
We start with:

**Dividend:**
\[ 4x^4 + 0x^3 + 4x - 10 \]

**Divisor:**
\[ 2x^2 - 3 \]

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

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

### Step 4: Repeat the Process
4. Now we take \( 6x^2 + 4x - 10 \) and divide by \( 2x^2 - 3 \):
\[
\frac{6x^2}{2x^2} = 3
\]
Thus, our next term in the quotient is \( 3 \).

5. Multiply the divisor by \( 3 \):
\[
3(2x^2 - 3) = 6x^2 - 9
\]
6. Subtract this from \( 6x^2 + 4x - 10 \):
\[
(6x^2 + 4x - 10) - (6x^2 - 9) = 4x - 1
\]

### Final Result
We now have:
- **Quotient:** \( 2x^2 + 3 \)
- **Remainder:** \( 4x - 1 \)

After reviewing the options given:
- The quotient is \( 2x^2 + 3 \), which corresponds to **Option A**.
- The remainder \( 4x - 1 \) corresponds to **Option B**.

So, the conclusion remains that:
- The quotient is \( 2x^2 + 3 \).
- The remainder is \( 4x - 1 \).

I apologize if there was any confusion in the prior explanation, but the calculations and the resulting options are accurate. Thank you for your patience!