Question

Divide using long division. State the​ quotient, q(x), and the​ remainder, r(x).
StartFraction 2 x cubed plus 10 x squared plus 15 x minus 18 Over x plus 2 EndFraction
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Part 1
StartFraction 2 x cubed plus 10 x squared plus 15 x minus 18 Over x plus 2 EndFraction
equals
  
enter your response hereplusStartFraction nothing Over x plus 2 EndFraction

Answer 1

To divide \(2x^3 + 10x^2 + 15x - 18\) by \(x + 2\) using long division, follow these steps:

1. **Divide the leading term:** Divide the leading term of the dividend \(2x^3\) by the leading term of the divisor \(x\):
\[
\frac{2x^3}{x} = 2x^2
\]
This is the first term of the quotient \(q(x)\).

2. **Multiply and subtract:** Multiply \(2x^2\) by the entire divisor \(x + 2\):
\[
2x^2(x + 2) = 2x^3 + 4x^2
\]
Subtract this from the original polynomial:
\[
(2x^3 + 10x^2 + 15x - 18) - (2x^3 + 4x^2) = 6x^2 + 15x - 18
\]

3. **Repeat steps:** Now, divide the new leading term \(6x^2\) by \(x\):
\[
\frac{6x^2}{x} = 6x
\]
Multiply \(6x\) by the divisor:
\[
6x(x + 2) = 6x^2 + 12x
\]
Subtract:
\[
(6x^2 + 15x - 18) - (6x^2 + 12x) = 3x - 18
\]

4. **Continue the process:** Divide \(3x\) by \(x\):
\[
\frac{3x}{x} = 3
\]
Multiply \(3\) by \(x + 2\):
\[
3(x + 2) = 3x + 6
\]
Subtract:
\[
(3x - 18) - (3x + 6) = -24
\]

5. **Write the division result:** The quotient is obtained from the terms we found:
\[
q(x) = 2x^2 + 6x + 3
\]
The remainder after the last subtraction is \(-24\).

Thus, writing the result in the specified format:
\[
\frac{2x^3 + 10x^2 + 15x - 18}{x + 2} = 2x^2 + 6x + 3 + \frac{-24}{x + 2}
\]

The final answer is:
- **Quotient:** \(q(x) = 2x^2 + 6x + 3\)
- **Remainder:** \(r(x) = -24\)

So you can express the division result as:
\[
\boxed{2x^2 + 6x + 3 + \frac{-24}{x + 2}}
\]