You divide p(x) in 3 term because it is large polynomial can I do it at one, i know it's look messy and result is same but can you do it for me please.

Sure, let's perform the polynomial long division of \( p(x) \) by \( g(x) \) in one continuous process:

1. **Divide the leading term of \( p(x) \) by the leading term of \( g(x) \)**:
   \[
   \frac{x^6}{x^2} = x^4
   \]

2. **Multiply \( x^4 \) by \( g(x) \)**:
   \[
   x^4(x^2 - 6x + 9) = x^6 - 6x^5 + 9x^4
   \]

3. **Subtract from \( p(x) \)**:
   \[
   (x^6 - 6x^5 + 17x^4 - 48x^3 + 76x^2 - 24x + 36) - (x^6 - 6x^5 + 9x^4) = 8x^4 - 48x^3 + 76x^2 - 24x + 36
   \]

4. **Next term**:
   - Divide \( 8x^4 \) by \( x^2 \): \( 8x^2 \).
   - Multiply \( 8x^2 \) by \( g(x) \): \( 8x^4 - 48x^3 + 72x^2 \).
   - Subtract: \( (8x^4 - 48x^3 + 76x^2 - 24x + 36) - (8x^4 - 48x^3 + 72x^2) = 4x^2 - 24x + 36 \).

5. **Final term**:
   - Divide \( 4x^2 \) by \( x^2 \): \( 4 \).
   - Multiply \( 4 \) by \( g(x) \): \( 4x^2 - 24x + 36 \).
   - Subtract: \( (4x^2 - 24x + 36) - (4x^2 - 24x + 36) = 0 \).

The quotient \( q(x) = x^4 + 8x^2 + 4 \) and the remainder \( r(x) = 0 \).

This confirms the quotient and remainder without breaking the process into parts. If you need further clarification or have any questions, feel free to ask!

Learn how to perform polynomial long division in one continuous process. This detailed guide covers dividing a polynomial by another polynomial, step-by-step, ensuring clarity and efficiency. Suitable for advanced high school students and anyone studying polynomial arithmetic.

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