what is the quotient and whats the remainder

To find the quotient and the remainder when \( P(x) = x^3 + 4x^2 + 7 \) is divided by \( x + 1 \), we can use polynomial long division or synthetic division. Here, I'll use synthetic division.

### Synthetic Division

1. **Set up the synthetic division.** Since we're dividing by \( x + 1 \), we use \( -1 \) (the root of \( x + 1 = 0 \)) for the synthetic division.

2. **Write the coefficients of the polynomial:**
   \[ 1 \quad 4 \quad 0 \quad 7 \]
   (Notice the 0 for the \( x \) term which is missing in the original polynomial.)

3. **Perform the synthetic division.**

```
   -1 |  1   4   0   7
      |     -1  -3   3
      ----------------
         1   3  -3  10
```

Here's how the calculations are done:
- Bring down the first coefficient (1).
- Multiply it by -1 and write the result under the next coefficient.
- Add the values in the second column: \( 4 + (-1) = 3 \).
- Multiply the result by -1 and write the result under the next coefficient.
- Add the values in the third column: \( 0 + (-3) = -3 \).
- Multiply the result by -1 and write the result under the next coefficient.
- Add the values in the fourth column: \( 7 + 3 = 10 \).

The bottom row gives the coefficients of the quotient polynomial and the remainder:
- Quotient: \( 1x^2 + 3x - 3 \)
- Remainder: \( 10 \)

Therefore, the quotient is \( x^2 + 3x - 3 \) and the remainder is \( 10 \).

Would you like any more details or have any questions?

Here are some related questions for further exploration:
1. How can you verify the result of the synthetic division?
2. How do you perform polynomial long division, and how does it compare to synthetic division?
3. Can you use synthetic division to find \( P(2) \) for \( P(x) = x^3 + 4x^2 + 7 \)?
4. What is the significance of the remainder in polynomial division?
5. How do you factorize the quotient \( x^2 + 3x - 3 \)?
6. What are the roots of the quotient polynomial \( x^2 + 3x - 3 \)?
7. How would you graph the polynomial \( P(x) = x^3 + 4x^2 + 7 \) including the quotient and remainder?
8. What happens if you divide \( P(x) = x^3 + 4x^2 + 7 \) by another polynomial, such as \( x - 2 \)?

**Tip:** Synthetic division is a quick and efficient method for dividing polynomials when the divisor is of the form \( x - a \).

Learn how to find the quotient and remainder when dividing the polynomial P(x) = x^3 + 4x^2 + 7 by x + 1 using synthetic division. Explore step-by-step calculations and understand the significance of quotient and remainder in polynomial division. Suitable for high school students studying polynomial division.

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