Answer :
Let's solve the problem of multiplying the two polynomials: [tex]\((x^4 + 1)\)[/tex] and [tex]\((3x^2 + 9x + 2)\)[/tex].
We'll do this by distributing each term from the first polynomial to each term in the second polynomial, combining like terms along the way.
1. Expand [tex]\((x^4 + 1)\)[/tex]:
- Multiply [tex]\(x^4\)[/tex] with every term in the second polynomial [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]
2. Expand [tex]\((1)\)[/tex]:
- Multiply [tex]\(1\)[/tex] with every term in the second polynomial [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]
3. Combine all the terms together:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
There's nothing to combine further, as all terms are of different degrees. Therefore, the final result of multiplying [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]
We'll do this by distributing each term from the first polynomial to each term in the second polynomial, combining like terms along the way.
1. Expand [tex]\((x^4 + 1)\)[/tex]:
- Multiply [tex]\(x^4\)[/tex] with every term in the second polynomial [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]
2. Expand [tex]\((1)\)[/tex]:
- Multiply [tex]\(1\)[/tex] with every term in the second polynomial [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]
3. Combine all the terms together:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
There's nothing to combine further, as all terms are of different degrees. Therefore, the final result of multiplying [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]
