the following given polynomials *P* (*x*) = 4*x*4 + 20*x*3 *−* 5*x*2 *−* 19*x* + 30 and *D*(*x*) = *x* + 5, use long or synthetic (only where appropriate) division to divide P(x) by D(x) in order to express the polynomial P in the form P(x) = D(x)Q(x) + R(x)

To divide \( P(x) = 4x^4 + 20x^3 - 5x^2 - 19x + 30 \) by \( D(x) = x + 5 \), we can use synthetic division since \( D(x) \) is a linear polynomial of the form \( x - c \) (here \( c = -5 \)).

### Step-by-Step Synthetic Division

1. **Set up the division**:  
   The divisor \( D(x) = x + 5 \) implies that \( c = -5 \).
   
2. **Write down the coefficients** of \( P(x) \):  
   \( 4, 20, -5, -19, 30 \)

3. **Perform synthetic division**:
   - **Step 1**: Bring down the first coefficient, \( 4 \).
   - **Step 2**: Multiply \( 4 \) by \( -5 \) to get \( -20 \), and add this to the next coefficient, \( 20 \), yielding \( 0 \).
   - **Step 3**: Multiply \( 0 \) by \( -5 \) to get \( 0 \), and add this to the next coefficient, \( -5 \), yielding \( -5 \).
   - **Step 4**: Multiply \( -5 \) by \( -5 \) to get \( 25 \), and add this to the next coefficient, \( -19 \), yielding \( 6 \).
   - **Step 5**: Multiply \( 6 \) by \( -5 \) to get \( -30 \), and add this to the last coefficient, \( 30 \), yielding \( 0 \).

The results of synthetic division are:

\[
4 \quad 0 \quad -5 \quad 6 \quad 0
\]

4. **Interpret the result**:
   - The quotient \( Q(x) = 4x^3 + 0x^2 - 5x + 6 \), or simply \( 4x^3 - 5x + 6 \).
   - The remainder \( R(x) = 0 \).

Thus, we can express \( P(x) \) as:

\[
P(x) = D(x) \cdot Q(x) + R(x) = (x + 5)(4x^3 - 5x + 6) + 0
\]

### Final Answer
\[
P(x) = (x + 5)(4x^3 - 5x + 6)
\]

Would you like further details on synthetic division, or do you have any questions?

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Here are some related questions to deepen understanding:
1. How would you perform long division instead of synthetic division on this problem?
2. Can synthetic division be applied to any polynomial divisor, or only certain types?
3. How can you verify that your division result is correct?
4. What happens if the remainder \( R(x) \) is not zero?
5. How do the degrees of \( P(x) \) and \( Q(x) \) relate to each other in polynomial division?

**Tip:** Synthetic division is a fast and efficient way to divide polynomials when dividing by a linear polynomial of the form \( x - c \).

The following given polynomials P(x) = 4x^4 + 20x^3 − 5x^2 − 19x + 30 and D(x) = x + 5, use long or synthetic (only where appropriate) division to divide P(x) by D(x) in order to express the polynomial P in the form P(x) = D(x)Q(x) + R(x)

Learn how to divide the polynomial P(x) = 4x^4 + 20x^3 − 5x^2 − 19x + 30 by D(x) = x + 5 using synthetic division. This solution provides a step-by-step approach to express P(x) in the form P(x) = D(x)Q(x) + R(x), covering essential algebraic concepts like polynomial division and the Remainder Theorem. Suitable for high school students in Grades 10-12.

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