Divide.

left parenthesis 4 x cubed minus 4 x squared minus 6 right parenthesis divided by left parenthesis x minus 3 right parenthesis4x3−4x2−6÷(x−3)

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Part 1

left parenthesis 4 x cubed minus 4 x squared minus 6 right parenthesis divided by left parenthesis x minus 3 right parenthesis4x3−4x2−6÷(x−3)equals=enter your response here 

​(Simplify your answer. Do not​ factor.)

To divide the polynomial \(4x^3 - 4x^2 - 6\) by \(x - 3\), we can use **polynomial long division**.

Here are the steps:

1. **Divide the leading term of the dividend** (the first term of \(4x^3 - 4x^2 - 6\)) by the leading term of the divisor \((x - 3)\). 

   \[
   \frac{4x^3}{x} = 4x^2
   \]

   Now, multiply \(4x^2\) by the divisor \((x - 3)\):

   \[
   4x^2 \cdot (x - 3) = 4x^3 - 12x^2
   \]

2. **Subtract** this result from the original polynomial:

   \[
   (4x^3 - 4x^2 - 6) - (4x^3 - 12x^2) = (4x^3 - 4x^2 - 6) - 4x^3 + 12x^2 = 8x^2 - 6
   \]

3. **Repeat the process** with the new dividend \(8x^2 - 6\). Divide the leading term \(8x^2\) by \(x\):

   \[
   \frac{8x^2}{x} = 8x
   \]

   Now, multiply \(8x\) by the divisor:

   \[
   8x \cdot (x - 3) = 8x^2 - 24x
   \]

4. **Subtract** this from the current dividend:

   \[
   (8x^2 - 6) - (8x^2 - 24x) = 8x^2 - 6 - 8x^2 + 24x = 24x - 6
   \]

5. **Repeat once more**, divide \(24x\) by \(x\):

   \[
   \frac{24x}{x} = 24
   \]

   Now, multiply \(24\) by the divisor:

   \[
   24 \cdot (x - 3) = 24x - 72
   \]

6. **Subtract**:

   \[
   (24x - 6) - (24x - 72) = 24x - 6 - 24x + 72 = 66
   \]

So, the result of the division is:

\[
4x^2 + 8x + 24 + \frac{66}{x - 3}
\]

### Final Answer:
\[
4x^2 + 8x + 24 + \frac{66}{x - 3}
\]

Would you like more details on polynomial long division, or do you have any questions?

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Here are five related questions to practice:

1. Divide \(6x^3 - 5x^2 + 4\) by \(x - 2\).
2. Perform the division: \(x^4 + 3x^3 - 2x\) by \(x + 1\).
3. Divide \(5x^2 - 7x + 9\) by \(x + 2\).
4. Divide \(8x^3 - 2x + 5\) by \(2x - 1\).
5. Simplify the expression \(2x^4 - 4x^3 + x - 5\) divided by \(x - 2\).

### Tip:
Always check for factoring opportunities before using long division; it can simplify the process!

left parenthesis 4 x cubed minus 4 x squared minus 6 right parenthesis divided by left parenthesis x minus 3 right parenthesis4x3−4x2−6÷(x−3)

Learn how to perform polynomial long division on 4x^3 - 4x^2 - 6 divided by (x - 3). This problem demonstrates key algebraic concepts, focusing on polynomial division using long division. Suitable for students in Grades 10-12.

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