To solve the problem of finding the polynomial equivalent to the expression [tex]\(\frac{x^2 + 6x + 9}{x + 3}\)[/tex], we can use polynomial long division.
### Step-by-Step Solution:
1. Set Up the Division:
- Dividend: [tex]\(x^2 + 6x + 9\)[/tex]
- Divisor: [tex]\(x + 3\)[/tex]
2. Divide the First Term:
- Divide the leading term of the dividend ([tex]\(x^2\)[/tex]) by the leading term of the divisor ([tex]\(x\)[/tex]):
[tex]\[
\frac{x^2}{x} = x
\][/tex]
- Place [tex]\(x\)[/tex] above the division line.
3. Multiply and Subtract:
- Multiply [tex]\(x\)[/tex] by the divisor ([tex]\(x + 3\)[/tex]):
[tex]\[
x \cdot (x + 3) = x^2 + 3x
\][/tex]
- Subtract this result from the original dividend:
[tex]\[
(x^2 + 6x + 9) - (x^2 + 3x) = 3x + 9
\][/tex]
4. Divide the Next Term:
- Divide the leading term of the new dividend ([tex]\(3x\)[/tex]) by the leading term of the divisor ([tex]\(x\)[/tex]):
[tex]\[
\frac{3x}{x} = 3
\][/tex]
- Place [tex]\(3\)[/tex] above the division line next to [tex]\(x\)[/tex].
5. Multiply and Subtract:
- Multiply [tex]\(3\)[/tex] by the divisor ([tex]\(x + 3\)[/tex]):
[tex]\[
3 \cdot (x + 3) = 3x + 9
\][/tex]
- Subtract this result from the current dividend:
[tex]\[
(3x + 9) - (3x + 9) = 0
\][/tex]
6. Conclusion:
- Since there is no remainder, the quotient of the division is [tex]\(x + 3\)[/tex].
Thus, the polynomial equivalent to the expression [tex]\(\frac{x^2 + 6x + 9}{x + 3}\)[/tex] is [tex]\(x + 3\)[/tex].
### Answer:
The correct answer is:
C. [tex]\(x + 3\)[/tex]
