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Answer :
To multiply the polynomials [tex]\((x+3)\)[/tex] and [tex]\((3x^2+8x+9)\)[/tex], we use the distributive property, also known as the FOIL method for binomials, but applied to a polynomial with more terms. Here's a step-by-step guide on how to do it:
1. Distribute [tex]\(x\)[/tex] to each term in the trinomial [tex]\((3x^2 + 8x + 9)\)[/tex]:
- Multiply [tex]\(x\)[/tex] by [tex]\(3x^2\)[/tex] to get [tex]\(3x^3\)[/tex].
- Multiply [tex]\(x\)[/tex] by [tex]\(8x\)[/tex] to get [tex]\(8x^2\)[/tex].
- Multiply [tex]\(x\)[/tex] by [tex]\(9\)[/tex] to get [tex]\(9x\)[/tex].
This results in: [tex]\(3x^3 + 8x^2 + 9x\)[/tex].
2. Distribute [tex]\(3\)[/tex] to each term in the trinomial [tex]\((3x^2 + 8x + 9)\)[/tex]:
- Multiply [tex]\(3\)[/tex] by [tex]\(3x^2\)[/tex] to get [tex]\(9x^2\)[/tex].
- Multiply [tex]\(3\)[/tex] by [tex]\(8x\)[/tex] to get [tex]\(24x\)[/tex].
- Multiply [tex]\(3\)[/tex] by [tex]\(9\)[/tex] to get [tex]\(27\)[/tex].
This results in: [tex]\(9x^2 + 24x + 27\)[/tex].
3. Combine like terms from both steps:
- 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 resulting expression is: [tex]\(3x^3 + 17x^2 + 33x + 27\)[/tex].
Therefore, the product of the polynomials [tex]\((x+3)\)[/tex] and [tex]\((3x^2+8x+9)\)[/tex] is [tex]\(3x^3 + 17x^2 + 33x + 27\)[/tex].
The correct answer is D. [tex]\(3x^3 + 17x^2 + 33x + 27\)[/tex].
1. Distribute [tex]\(x\)[/tex] to each term in the trinomial [tex]\((3x^2 + 8x + 9)\)[/tex]:
- Multiply [tex]\(x\)[/tex] by [tex]\(3x^2\)[/tex] to get [tex]\(3x^3\)[/tex].
- Multiply [tex]\(x\)[/tex] by [tex]\(8x\)[/tex] to get [tex]\(8x^2\)[/tex].
- Multiply [tex]\(x\)[/tex] by [tex]\(9\)[/tex] to get [tex]\(9x\)[/tex].
This results in: [tex]\(3x^3 + 8x^2 + 9x\)[/tex].
2. Distribute [tex]\(3\)[/tex] to each term in the trinomial [tex]\((3x^2 + 8x + 9)\)[/tex]:
- Multiply [tex]\(3\)[/tex] by [tex]\(3x^2\)[/tex] to get [tex]\(9x^2\)[/tex].
- Multiply [tex]\(3\)[/tex] by [tex]\(8x\)[/tex] to get [tex]\(24x\)[/tex].
- Multiply [tex]\(3\)[/tex] by [tex]\(9\)[/tex] to get [tex]\(27\)[/tex].
This results in: [tex]\(9x^2 + 24x + 27\)[/tex].
3. Combine like terms from both steps:
- 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 resulting expression is: [tex]\(3x^3 + 17x^2 + 33x + 27\)[/tex].
Therefore, the product of the polynomials [tex]\((x+3)\)[/tex] and [tex]\((3x^2+8x+9)\)[/tex] is [tex]\(3x^3 + 17x^2 + 33x + 27\)[/tex].
The correct answer is D. [tex]\(3x^3 + 17x^2 + 33x + 27\)[/tex].
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