Question

Consider the polynomial division (x^3+5x^2+4x+18)÷(x+5) . Give the quotient with the remainder expressed as a fraction.

Answer 1

To perform the polynomial division \((x^3 + 5x^2 + 4x + 18) \div (x + 5)\), we will divide the polynomial \(x^3 + 5x^2 + 4x + 18\) by \(x + 5\) using long division.

1. **Divide the leading term:**
- Divide the leading term \(x^3\) by the leading term \(x\):
\[
x^3 \div x = x^2
\]

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

3. **Repeat the process:**
- Now, divide the new leading term \(4x\) by \(x\):
\[
4x \div x = 4
\]
- Multiply \(4\) by \((x + 5)\):
\[
4(x + 5) = 4x + 20
\]
- Subtract this from \(4x + 18\):
\[
(4x + 18) - (4x + 20) = 18 - 20 = -2
\]

Now that we have completed the division, the quotient is:
\[
x^2 + 4
\]
And the remainder is:
\[
-2
\]

4. **Express the remainder as a fraction:**
Therefore, we can express the final result of the polynomial division as:
\[
x^2 + 4 - \frac{2}{x + 5}
\]

Combining all parts, the final quotient with the remainder expressed as a fraction is:
\[
\boxed{x^2 + 4 - \frac{2}{x + 5}}
\]