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Answer :
Sure! Let's multiply the two polynomials [tex]\((x^4 + 1)\)[/tex] and [tex]\((3x^2 + 9x + 2)\)[/tex] using the distributive property. We'll break it down step-by-step:
1. Distribute each term of the first polynomial to every term of the second polynomial:
- Multiply [tex]\(x^4\)[/tex] by each term in [tex]\((3x^2 + 9x + 2)\)[/tex]:
- [tex]\(x^4 \cdot 3x^2 = 3x^{6}\)[/tex]
- [tex]\(x^4 \cdot 9x = 9x^{5}\)[/tex]
- [tex]\(x^4 \cdot 2 = 2x^{4}\)[/tex]
- Multiply [tex]\(1\)[/tex] (the second term of the first polynomial) by each term in [tex]\((3x^2 + 9x + 2)\)[/tex]:
- [tex]\(1 \cdot 3x^2 = 3x^{2}\)[/tex]
- [tex]\(1 \cdot 9x = 9x\)[/tex]
- [tex]\(1 \cdot 2 = 2\)[/tex]
2. Combine all the results from the distributions:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
These terms are already in their simplest form, as there are no like terms to combine.
Therefore, the final expanded polynomial is:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
I hope this helps! Let me know if you have any other questions.
1. Distribute each term of the first polynomial to every term of the second polynomial:
- Multiply [tex]\(x^4\)[/tex] by each term in [tex]\((3x^2 + 9x + 2)\)[/tex]:
- [tex]\(x^4 \cdot 3x^2 = 3x^{6}\)[/tex]
- [tex]\(x^4 \cdot 9x = 9x^{5}\)[/tex]
- [tex]\(x^4 \cdot 2 = 2x^{4}\)[/tex]
- Multiply [tex]\(1\)[/tex] (the second term of the first polynomial) by each term in [tex]\((3x^2 + 9x + 2)\)[/tex]:
- [tex]\(1 \cdot 3x^2 = 3x^{2}\)[/tex]
- [tex]\(1 \cdot 9x = 9x\)[/tex]
- [tex]\(1 \cdot 2 = 2\)[/tex]
2. Combine all the results from the distributions:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
These terms are already in their simplest form, as there are no like terms to combine.
Therefore, the final expanded polynomial is:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
I hope this helps! Let me know if you have any other questions.
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