Question

consider the polynomial division (x^3+5x^2+4x+18) divided by (x+5) give the quotient with the remainder expressed as a fraction

Answer 1

To perform the polynomial division of \( x^3 + 5x^2 + 4x + 18 \) by \( x + 5 \), we can follow the polynomial long division method.

1. **Divide the leading term**:
- Divide the leading term of the dividend \( x^3 \) by the leading term of the divisor \( x \) to get \( x^2 \).

2. **Multiply**:
- Multiply \( x^2 \) by \( x + 5 \):
\[
x^2(x + 5) = x^3 + 5x^2
\]

3. **Subtract**:
- Subtract this from the original polynomial:
\[
(x^3 + 5x^2 + 4x + 18) - (x^3 + 5x^2) = (0 + 4x + 18) = 4x + 18
\]

4. **Repeat the process**:
- Divide the leading term of the new polynomial \( 4x \) by the leading term of the divisor \( x \) to get \( 4 \).
- Multiply \( 4 \) by \( x + 5 \):
\[
4(x + 5) = 4x + 20
\]

5. **Subtract**:
- Now subtract this from \( 4x + 18 \):
\[
(4x + 18) - (4x + 20) = 18 - 20 = -2
\]

Now, we have reached a result. The division can be summarized as follows:

- Quotient: \( x^2 + 4 \)
- Remainder: \( -2 \)

Thus, we can express the result of the division as:
\[
\frac{x^3 + 5x^2 + 4x + 18}{x + 5} = x^2 + 4 + \frac{-2}{x + 5}
\]

or simply:

\[
x^2 + 4 - \frac{2}{x + 5}
\]

So the final answer is:
- **Quotient**: \( x^2 + 4 \)
- **Remainder**: \( -2 \), expressed as a fraction: \( \frac{-2}{x + 5} \)