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Answer :
Let's multiply the polynomials [tex]\((x^4 + 1)\)[/tex] and [tex]\((3x^2 + 9x + 2)\)[/tex] step-by-step:
1. Distribute each term in the first polynomial to every term in the second polynomial:
- Multiply [tex]\(x^4\)[/tex] by each term in the second polynomial:
[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] by each term in the second polynomial:
[tex]\[
1 \cdot (3x^2) = 3x^2
\][/tex]
[tex]\[
1 \cdot (9x) = 9x
\][/tex]
[tex]\[
1 \cdot 2 = 2
\][/tex]
2. Combine all these products:
The results from the distribution step are:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
3. Write the final expression:
The resulting polynomial after multiplication is:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
Therefore, the answer is [tex]\(3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2\)[/tex].
1. Distribute each term in the first polynomial to every term in the second polynomial:
- Multiply [tex]\(x^4\)[/tex] by each term in the second polynomial:
[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] by each term in the second polynomial:
[tex]\[
1 \cdot (3x^2) = 3x^2
\][/tex]
[tex]\[
1 \cdot (9x) = 9x
\][/tex]
[tex]\[
1 \cdot 2 = 2
\][/tex]
2. Combine all these products:
The results from the distribution step are:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
3. Write the final expression:
The resulting polynomial after multiplication is:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
Therefore, the answer is [tex]\(3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2\)[/tex].
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