Answer :
Let's multiply the polynomials [tex]\((x + 3)\)[/tex] and [tex]\((3x^2 + 8x + 9)\)[/tex] step-by-step.
1. Distribute the [tex]\(x\)[/tex]:
- Multiply [tex]\(x\)[/tex] by each term in the second polynomial:
- [tex]\(x \times 3x^2 = 3x^3\)[/tex]
- [tex]\(x \times 8x = 8x^2\)[/tex]
- [tex]\(x \times 9 = 9x\)[/tex]
So, the result from the first distribution is:
[tex]\[
3x^3 + 8x^2 + 9x
\][/tex]
2. Distribute the [tex]\(3\)[/tex]:
- Multiply [tex]\(3\)[/tex] by each term in the second polynomial:
- [tex]\(3 \times 3x^2 = 9x^2\)[/tex]
- [tex]\(3 \times 8x = 24x\)[/tex]
- [tex]\(3 \times 9 = 27\)[/tex]
So, the result from this distribution is:
[tex]\[
9x^2 + 24x + 27
\][/tex]
3. Combine the like terms:
Now, add the results from both distributions:
[tex]\[
3x^3 + 8x^2 + 9x + 9x^2 + 24x + 27
\][/tex]
Combine like terms:
- The [tex]\(x^3\)[/tex] term is [tex]\(3x^3\)[/tex].
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(8x^2 + 9x^2 = 17x^2\)[/tex].
- Combine the [tex]\(x\)[/tex] terms: [tex]\(9x + 24x = 33x\)[/tex].
- The constant term is [tex]\(27\)[/tex].
Therefore, the final polynomial is:
[tex]\[
3x^3 + 17x^2 + 33x + 27
\][/tex]
Thus, the correct answer is:
D. [tex]\(3x^3 + 17x^2 + 33x + 27\)[/tex]
1. Distribute the [tex]\(x\)[/tex]:
- Multiply [tex]\(x\)[/tex] by each term in the second polynomial:
- [tex]\(x \times 3x^2 = 3x^3\)[/tex]
- [tex]\(x \times 8x = 8x^2\)[/tex]
- [tex]\(x \times 9 = 9x\)[/tex]
So, the result from the first distribution is:
[tex]\[
3x^3 + 8x^2 + 9x
\][/tex]
2. Distribute the [tex]\(3\)[/tex]:
- Multiply [tex]\(3\)[/tex] by each term in the second polynomial:
- [tex]\(3 \times 3x^2 = 9x^2\)[/tex]
- [tex]\(3 \times 8x = 24x\)[/tex]
- [tex]\(3 \times 9 = 27\)[/tex]
So, the result from this distribution is:
[tex]\[
9x^2 + 24x + 27
\][/tex]
3. Combine the like terms:
Now, add the results from both distributions:
[tex]\[
3x^3 + 8x^2 + 9x + 9x^2 + 24x + 27
\][/tex]
Combine like terms:
- The [tex]\(x^3\)[/tex] term is [tex]\(3x^3\)[/tex].
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(8x^2 + 9x^2 = 17x^2\)[/tex].
- Combine the [tex]\(x\)[/tex] terms: [tex]\(9x + 24x = 33x\)[/tex].
- The constant term is [tex]\(27\)[/tex].
Therefore, the final polynomial is:
[tex]\[
3x^3 + 17x^2 + 33x + 27
\][/tex]
Thus, the correct answer is:
D. [tex]\(3x^3 + 17x^2 + 33x + 27\)[/tex]
