Divide the product of `(4x^(2)-9)` and `(2x^(2)-3x+1)` by `(4x^(3)-7x+3)` .

Options

A. `2x-3`
B. `2x+3`
C. `2x`
D. `3x-2`

Answer

To solve the problem of dividing the product of \( (4x^2 - 9) \) and \( (2x^2 - 3x + 1) \) by \( (4x^3 - 7x + 3) \), we will follow these steps:

### Step 1: Expand the Product
First, we need to find the product of \( (4x^2 - 9) \) and \( (2x^2 - 3x + 1) \).

\[ (4x^2 - 9)(2x^2 - 3x + 1) \]

Using the distributive property (FOIL method):

\[ = 4x^2 \cdot 2x^2 + 4x^2 \cdot (-3x) + 4x^2 \cdot 1 - 9 \cdot 2x^2 + (-9) \cdot (-3x) + (-9) \cdot 1 \]

Calculating each term:

\[ = 8x^4 - 12x^3 + 4x^2 - 18x^2 + 27x - 9 \]

Now, combine like terms:

\[ = 8x^4 - 12x^3 + (4x^2 - 18x^2) + 27x - 9 \]
\[ = 8x^4 - 12x^3 - 14x^2 + 27x - 9 \]

### Step 2: Set Up the Division
Now we will divide the expanded polynomial \( 8x^4 - 12x^3 - 14x^2 + 27x - 9 \) by \( 4x^3 - 7x + 3 \).

### Step 3: Perform Polynomial Long Division
1. Divide the leading term of the dividend by the leading term of the divisor:
\[ \frac{8x^4}{4x^3} = 2x \]

2. Multiply the entire divisor by \( 2x \):
\[ 2x(4x^3 - 7x + 3) = 8x^4 - 14x^2 + 6x \]

3. Subtract this from the original polynomial:
\[ (8x^4 - 12x^3 - 14x^2 + 27x - 9) - (8x^4 - 14x^2 + 6x) \]
This simplifies to:
\[ -12x^3 + 0x^2 + (27x - 6x) - 9 = -12x^3 + 21x - 9 \]

4. Now, repeat the process:
- Divide the leading term:
\[ \frac{-12x^3}{4x^3} = -3 \]

5. Multiply the entire divisor by \(-3\):
\[ -3(4x^3 - 7x + 3) = -12x^3 + 21x - 9 \]

6. Subtract again:
\[ (-12x^3 + 21x - 9) - (-12x^3 + 21x - 9) = 0 \]

### Final Result
The result of the division is:
\[ 2x - 3 \]