Two polynomials P and D are given. Use either synthetic or long division to divide 
P(x)
 by 
D(x),
 and express the quotient 
P(x)/D(x)
 in the form 
P(x)
D(x)
 = Q(x) + 
R(x)
D(x)
.
P(x) = 4x4 − 2x3 + 20x2,    D(x) = 2x2 + 9

We are tasked with dividing the polynomial \( P(x) = 4x^4 - 2x^3 + 20x^2 \) by \( D(x) = 2x^2 + 9 \) using either synthetic or long division, and expressing the result in the form:

\[
\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}
\]

### Step 1: Setting up for Long Division
We’ll use long division for this problem. Write the terms of both \( P(x) \) and \( D(x) \) in standard form. 

#### Dividend: \( P(x) = 4x^4 - 2x^3 + 20x^2 \)
Notice that there are no \(x\) or constant terms, so we can rewrite it as:
\[
P(x) = 4x^4 - 2x^3 + 20x^2 + 0x + 0
\]

#### Divisor: \( D(x) = 2x^2 + 9 \)

### Step 2: Perform the Division

1. **Divide the first term of \( P(x) \) by the first term of \( D(x) \):**

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

   So, the first term of the quotient \( Q(x) \) is \( 2x^2 \).

2. **Multiply \( 2x^2 \) by \( D(x) \):**
   
   \[
   2x^2 \times (2x^2 + 9) = 4x^4 + 18x^2
   \]

3. **Subtract this product from \( P(x) \):**

   \[
   (4x^4 - 2x^3 + 20x^2) - (4x^4 + 0x^3 + 18x^2) = -2x^3 + 2x^2
   \]

4. **Divide the first term of the new polynomial by the first term of \( D(x) \):**

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

   So, the next term of \( Q(x) \) is \( -x \).

5. **Multiply \( -x \) by \( D(x) \):**

   \[
   -x \times (2x^2 + 9) = -2x^3 - 9x
   \]

6. **Subtract this product from the current polynomial:**

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

7. **Divide the first term of the new polynomial by the first term of \( D(x) \):**

   \[
   \frac{2x^2}{2x^2} = 1
   \]

   So, the next term of \( Q(x) \) is \( 1 \).

8. **Multiply \( 1 \) by \( D(x) \):**

   \[
   1 \times (2x^2 + 9) = 2x^2 + 9
   \]

9. **Subtract this product from the current polynomial:**

   \[
   (2x^2 + 9x) - (2x^2 + 9) = 9x - 9
   \]

Now we have the remainder \( R(x) = 9x - 9 \).

### Step 3: Writing the Final Answer

The quotient \( Q(x) \) is \( 2x^2 - x + 1 \), and the remainder is \( R(x) = 9x - 9 \). Therefore, the division of \( P(x) \) by \( D(x) \) can be written as:

\[
\frac{P(x)}{D(x)} = 2x^2 - x + 1 + \frac{9x - 9}{2x^2 + 9}
\]

Would you like further details or clarifications?

### Related Questions:
1. How can synthetic division be used for dividing polynomials?
2. What are the advantages of using long division over synthetic division?
3. How do you find the quotient and remainder when dividing two polynomials?
4. Can polynomial division result in a quotient without a remainder?
5. What is the significance of the degree of the divisor in polynomial division?

### Tip:
Always align the terms by degree when performing long division to avoid mistakes.

Two polynomials P and D are given. Use either synthetic or long division to divide P(x) by D(x), and express the quotient P(x)/D(x) in the form P(x)/D(x) = Q(x) + R(x)/D(x). P(x) = 4x^4 − 2x^3 + 20x^2, D(x) = 2x^2 + 9

Solve the polynomial division of P(x) = 4x^4 − 2x^3 + 20x^2 by D(x) = 2x^2 + 9 using long division. This detailed step-by-step solution covers polynomial division concepts and expressions for the quotient and remainder. Suitable for students in Grades 10-12.

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