To solve the given problem, we need to perform polynomial division for each expression and find the corresponding quotients. Let's match them step-by-step based on the pre-determined results.
### Step-by-Step Breakdown
1. Expression 1:
[tex]\[
\left(x^2 - 3x - 18\right) \div (x - 6)
\][/tex]
- Quotient: By dividing [tex]\(x^2 - 3x - 18\)[/tex] by [tex]\(x - 6\)[/tex], we obtain:
[tex]\[
x + 3
\][/tex]
- Remainder: The remainder is [tex]\(0\)[/tex].
2. Expression 2:
[tex]\[
\left(x^3 - x^2 - 5x - 3\right) \div \left(x^2 + 2x + 1\right)
\][/tex]
- Quotient: By dividing [tex]\(x^3 - x^2 - 5x - 3\)[/tex] by [tex]\(x^2 + 2x + 1\)[/tex], we obtain:
[tex]\[
x - 3
\][/tex]
- Remainder: The remainder is [tex]\(0\)[/tex].
3. Expression 3:
[tex]\[
\left(x^3 + 2x^2 - 1\right) \div \left(x^2 + x + 1\right)
\][/tex]
- Quotient: By dividing [tex]\(x^3 + 2x^2 - 1\)[/tex] by [tex]\(x^2 + x + 1\)[/tex], we obtain:
[tex]\[
x + 1
\][/tex]
- Remainder: The remainder is [tex]\(-2x - 2\)[/tex], which is often denoted as [tex]\( x + 1 \, R \, -2x - 2 \)[/tex].
### Matching Expressions and Quotients
Now, we match each expression with its quotient:
- [tex]\(\left(x^2 - 3x - 18 \right) \div (x - 6)\)[/tex] matches [tex]\(x + 3\)[/tex].
- [tex]\(\left(x^3 - x^2 - 5x - 3 \right) \div \left(x^2 + 2x + 1\right)\)[/tex] matches [tex]\(x - 3\)[/tex].
- [tex]\(\left(x^3 + 2x^2 - 1\right) \div \left(x^2 + x + 1\right)\)[/tex] matches [tex]\(x + 1 \, R \, -2x - 2\)[/tex].
Therefore, the final matches are:
1. [tex]\(\left(x^2 - 3x - 18\right) \div (x - 6)\)[/tex] → [tex]\(x + 3\)[/tex]
2. [tex]\(\left(x^3 - x^2 - 5x - 3\right) \div \left(x^2 + 2x + 1\right)\)[/tex] → [tex]\(x - 3\)[/tex]
3. [tex]\(\left(x^3 + 2x^2 - 1\right) \div \left(x^2 + x + 1\right)\)[/tex] → [tex]\(x + 1 \, R \, -2x - 2\)[/tex]
This ensures that each polynomial division is correctly matched with its quotient and remainder.
