Answer :
To solve the problem of multiplying the polynomials [tex]\((x^4 + 1)\)[/tex] and [tex]\((3x^2 + 9x + 2)\)[/tex], follow these steps:
1. Distribute each term of the first polynomial to every term of the second polynomial:
- Multiply [tex]\(x^4\)[/tex] by each term in the second polynomial:
- [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 in the second polynomial:
- [tex]\(1 \times 3x^2 = 3x^2\)[/tex]
- [tex]\(1 \times 9x = 9x\)[/tex]
- [tex]\(1 \times 2 = 2\)[/tex]
2. Combine all the results:
After distributing, you add all the results together to get the final expanded polynomial:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
So, the result of multiplying the polynomials [tex]\((x^4 + 1)\)[/tex] and [tex]\((3x^2 + 9x + 2)\)[/tex] is [tex]\(3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2\)[/tex].
1. Distribute each term of the first polynomial to every term of the second polynomial:
- Multiply [tex]\(x^4\)[/tex] by each term in the second polynomial:
- [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 in the second polynomial:
- [tex]\(1 \times 3x^2 = 3x^2\)[/tex]
- [tex]\(1 \times 9x = 9x\)[/tex]
- [tex]\(1 \times 2 = 2\)[/tex]
2. Combine all the results:
After distributing, you add all the results together to get the final expanded polynomial:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
So, the result of multiplying the polynomials [tex]\((x^4 + 1)\)[/tex] and [tex]\((3x^2 + 9x + 2)\)[/tex] is [tex]\(3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2\)[/tex].
