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Answer :
To multiply the polynomials [tex]\((x+3)\)[/tex] and [tex]\((3x^2+8x+9)\)[/tex], you need to distribute each term in the first polynomial by each term in the second polynomial and then combine like terms. Here’s how you can do it step-by-step:
1. Distribute [tex]\(x\)[/tex] from [tex]\((x+3)\)[/tex]:
- Multiply [tex]\(x\)[/tex] by [tex]\(3x^2\)[/tex]:
[tex]\[
x \cdot 3x^2 = 3x^3
\][/tex]
- Multiply [tex]\(x\)[/tex] by [tex]\(8x\)[/tex]:
[tex]\[
x \cdot 8x = 8x^2
\][/tex]
- Multiply [tex]\(x\)[/tex] by [tex]\(9\)[/tex]:
[tex]\[
x \cdot 9 = 9x
\][/tex]
2. Distribute [tex]\(3\)[/tex] from [tex]\((x+3)\)[/tex]:
- Multiply [tex]\(3\)[/tex] by [tex]\(3x^2\)[/tex]:
[tex]\[
3 \cdot 3x^2 = 9x^2
\][/tex]
- Multiply [tex]\(3\)[/tex] by [tex]\(8x\)[/tex]:
[tex]\[
3 \cdot 8x = 24x
\][/tex]
- Multiply [tex]\(3\)[/tex] by [tex]\(9\)[/tex]:
[tex]\[
3 \cdot 9 = 27
\][/tex]
3. Combine like terms:
- Combine [tex]\(x^3\)[/tex] terms:
[tex]\[
3x^3
\][/tex]
- Combine [tex]\(x^2\)[/tex] terms:
[tex]\[
8x^2 + 9x^2 = 17x^2
\][/tex]
- Combine [tex]\(x\)[/tex] terms:
[tex]\[
9x + 24x = 33x
\][/tex]
- The constant term is [tex]\(27\)[/tex].
Putting it all together, the product of the polynomials is:
[tex]\[
3x^3 + 17x^2 + 33x + 27
\][/tex]
Therefore, the correct answer is option D: [tex]\(3x^3 + 17x^2 + 33x + 27\)[/tex].
1. Distribute [tex]\(x\)[/tex] from [tex]\((x+3)\)[/tex]:
- Multiply [tex]\(x\)[/tex] by [tex]\(3x^2\)[/tex]:
[tex]\[
x \cdot 3x^2 = 3x^3
\][/tex]
- Multiply [tex]\(x\)[/tex] by [tex]\(8x\)[/tex]:
[tex]\[
x \cdot 8x = 8x^2
\][/tex]
- Multiply [tex]\(x\)[/tex] by [tex]\(9\)[/tex]:
[tex]\[
x \cdot 9 = 9x
\][/tex]
2. Distribute [tex]\(3\)[/tex] from [tex]\((x+3)\)[/tex]:
- Multiply [tex]\(3\)[/tex] by [tex]\(3x^2\)[/tex]:
[tex]\[
3 \cdot 3x^2 = 9x^2
\][/tex]
- Multiply [tex]\(3\)[/tex] by [tex]\(8x\)[/tex]:
[tex]\[
3 \cdot 8x = 24x
\][/tex]
- Multiply [tex]\(3\)[/tex] by [tex]\(9\)[/tex]:
[tex]\[
3 \cdot 9 = 27
\][/tex]
3. Combine like terms:
- Combine [tex]\(x^3\)[/tex] terms:
[tex]\[
3x^3
\][/tex]
- Combine [tex]\(x^2\)[/tex] terms:
[tex]\[
8x^2 + 9x^2 = 17x^2
\][/tex]
- Combine [tex]\(x\)[/tex] terms:
[tex]\[
9x + 24x = 33x
\][/tex]
- The constant term is [tex]\(27\)[/tex].
Putting it all together, the product of the polynomials is:
[tex]\[
3x^3 + 17x^2 + 33x + 27
\][/tex]
Therefore, the correct answer is option D: [tex]\(3x^3 + 17x^2 + 33x + 27\)[/tex].
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