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Answer :
To solve the problem of multiplying [tex]\((x^4 + 1)(3x^2 + 9x + 2)\)[/tex] with [tex]\((x^4 + 3x^2 + 9x + 3)\)[/tex], let's break it down step by step using distributive property and polynomial multiplication:
1. First Step: Expand [tex]\((x^4 + 1)(3x^2 + 9x + 2)\)[/tex].
- Distribute each term in [tex]\(x^4 + 1\)[/tex] to the terms in [tex]\(3x^2 + 9x + 2\)[/tex]:
- Multiply [tex]\(x^4\)[/tex] by each term:
- [tex]\(x^4 \times 3x^2 = 3x^6\)[/tex]
- [tex]\(x^4 \times 9x = 9x^5\)[/tex]
- [tex]\(x^4 \times 2 = 2x^4\)[/tex]
- Multiply [tex]\(1\)[/tex] by each term:
- [tex]\(1 \times 3x^2 = 3x^2\)[/tex]
- [tex]\(1 \times 9x = 9x\)[/tex]
- [tex]\(1 \times 2 = 2\)[/tex]
- Combine the results:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
2. Second Step: Expand the resulting expression with [tex]\((x^4 + 3x^2 + 9x + 3)\)[/tex].
- Distribute each term from the first multiplication result to each term in the second expression:
- Start with [tex]\(3x^6\)[/tex]:
- [tex]\(3x^6 \times x^4 = 3x^{10}\)[/tex]
- [tex]\(3x^6 \times 3x^2 = 9x^8\)[/tex]
- [tex]\(3x^6 \times 9x = 27x^7\)[/tex]
- [tex]\(3x^6 \times 3 = 9x^6\)[/tex]
- Next with [tex]\(9x^5\)[/tex]:
- [tex]\(9x^5 \times x^4 = 9x^9\)[/tex]
- [tex]\(9x^5 \times 3x^2 = 27x^7\)[/tex]
- [tex]\(9x^5 \times 9x = 81x^6\)[/tex]
- [tex]\(9x^5 \times 3 = 27x^5\)[/tex]
- Continue with [tex]\(2x^4\)[/tex]:
- [tex]\(2x^4 \times x^4 = 2x^8\)[/tex]
- [tex]\(2x^4 \times 3x^2 = 6x^6\)[/tex]
- [tex]\(2x^4 \times 9x = 18x^5\)[/tex]
- [tex]\(2x^4 \times 3 = 6x^4\)[/tex]
- Then with [tex]\(3x^2\)[/tex]:
- [tex]\(3x^2 \times x^4 = 3x^6\)[/tex]
- [tex]\(3x^2 \times 3x^2 = 9x^4\)[/tex]
- [tex]\(3x^2 \times 9x = 27x^3\)[/tex]
- [tex]\(3x^2 \times 3 = 9x^2\)[/tex]
- Now with [tex]\(9x\)[/tex]:
- [tex]\(9x \times x^4 = 9x^5\)[/tex]
- [tex]\(9x \times 3x^2 = 27x^3\)[/tex]
- [tex]\(9x \times 9x = 81x^2\)[/tex]
- [tex]\(9x \times 3 = 27x\)[/tex]
- Finally with [tex]\(2\)[/tex]:
- [tex]\(2 \times x^4 = 2x^4\)[/tex]
- [tex]\(2 \times 3x^2 = 6x^2\)[/tex]
- [tex]\(2 \times 9x = 18x\)[/tex]
- [tex]\(2 \times 3 = 6\)[/tex]
3. Combine and Simplify Terms:
- Carefully add all the terms together, combining like terms to reach the final polynomial expression:
[tex]\[
3x^{10} + 9x^9 + 11x^8 + 54x^7 + 99x^6 + 54x^5 + 17x^4 + 54x^3 + 96x^2 + 45x + 6
\][/tex]
This final expression is the result of multiplying the given polynomials.
1. First Step: Expand [tex]\((x^4 + 1)(3x^2 + 9x + 2)\)[/tex].
- Distribute each term in [tex]\(x^4 + 1\)[/tex] to the terms in [tex]\(3x^2 + 9x + 2\)[/tex]:
- Multiply [tex]\(x^4\)[/tex] by each term:
- [tex]\(x^4 \times 3x^2 = 3x^6\)[/tex]
- [tex]\(x^4 \times 9x = 9x^5\)[/tex]
- [tex]\(x^4 \times 2 = 2x^4\)[/tex]
- Multiply [tex]\(1\)[/tex] by each term:
- [tex]\(1 \times 3x^2 = 3x^2\)[/tex]
- [tex]\(1 \times 9x = 9x\)[/tex]
- [tex]\(1 \times 2 = 2\)[/tex]
- Combine the results:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
2. Second Step: Expand the resulting expression with [tex]\((x^4 + 3x^2 + 9x + 3)\)[/tex].
- Distribute each term from the first multiplication result to each term in the second expression:
- Start with [tex]\(3x^6\)[/tex]:
- [tex]\(3x^6 \times x^4 = 3x^{10}\)[/tex]
- [tex]\(3x^6 \times 3x^2 = 9x^8\)[/tex]
- [tex]\(3x^6 \times 9x = 27x^7\)[/tex]
- [tex]\(3x^6 \times 3 = 9x^6\)[/tex]
- Next with [tex]\(9x^5\)[/tex]:
- [tex]\(9x^5 \times x^4 = 9x^9\)[/tex]
- [tex]\(9x^5 \times 3x^2 = 27x^7\)[/tex]
- [tex]\(9x^5 \times 9x = 81x^6\)[/tex]
- [tex]\(9x^5 \times 3 = 27x^5\)[/tex]
- Continue with [tex]\(2x^4\)[/tex]:
- [tex]\(2x^4 \times x^4 = 2x^8\)[/tex]
- [tex]\(2x^4 \times 3x^2 = 6x^6\)[/tex]
- [tex]\(2x^4 \times 9x = 18x^5\)[/tex]
- [tex]\(2x^4 \times 3 = 6x^4\)[/tex]
- Then with [tex]\(3x^2\)[/tex]:
- [tex]\(3x^2 \times x^4 = 3x^6\)[/tex]
- [tex]\(3x^2 \times 3x^2 = 9x^4\)[/tex]
- [tex]\(3x^2 \times 9x = 27x^3\)[/tex]
- [tex]\(3x^2 \times 3 = 9x^2\)[/tex]
- Now with [tex]\(9x\)[/tex]:
- [tex]\(9x \times x^4 = 9x^5\)[/tex]
- [tex]\(9x \times 3x^2 = 27x^3\)[/tex]
- [tex]\(9x \times 9x = 81x^2\)[/tex]
- [tex]\(9x \times 3 = 27x\)[/tex]
- Finally with [tex]\(2\)[/tex]:
- [tex]\(2 \times x^4 = 2x^4\)[/tex]
- [tex]\(2 \times 3x^2 = 6x^2\)[/tex]
- [tex]\(2 \times 9x = 18x\)[/tex]
- [tex]\(2 \times 3 = 6\)[/tex]
3. Combine and Simplify Terms:
- Carefully add all the terms together, combining like terms to reach the final polynomial expression:
[tex]\[
3x^{10} + 9x^9 + 11x^8 + 54x^7 + 99x^6 + 54x^5 + 17x^4 + 54x^3 + 96x^2 + 45x + 6
\][/tex]
This final expression is the result of multiplying the given polynomials.
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