Answer :
To solve the multiplication of the polynomials [tex]\((x^4 + 1)\)[/tex] and [tex]\((3x^2 + 9x + 2)\)[/tex], we will expand the expression step-by-step using the distributive property.
1. Expand each term of [tex]\( x^4 + 1 \)[/tex] with the terms in [tex]\( 3x^2 + 9x + 2 \)[/tex]:
- Start with [tex]\( x^4 \)[/tex]:
- Multiply [tex]\( x^4 \)[/tex] by each term in [tex]\((3x^2 + 9x + 2)\)[/tex]:
[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]
- Now expand 1 in the same way:
- Multiply 1 by each term in [tex]\((3x^2 + 9x + 2)\)[/tex]:
[tex]\[
1 \times 3x^2 = 3x^2
\][/tex]
[tex]\[
1 \times 9x = 9x
\][/tex]
[tex]\[
1 \times 2 = 2
\][/tex]
2. Combine all the products we've obtained:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
After expanding and combining like terms, the final expanded polynomial is:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
This is the result of multiplying [tex]\((x^4 + 1)\)[/tex] by [tex]\((3x^2 + 9x + 2)\)[/tex].
1. Expand each term of [tex]\( x^4 + 1 \)[/tex] with the terms in [tex]\( 3x^2 + 9x + 2 \)[/tex]:
- Start with [tex]\( x^4 \)[/tex]:
- Multiply [tex]\( x^4 \)[/tex] by each term in [tex]\((3x^2 + 9x + 2)\)[/tex]:
[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]
- Now expand 1 in the same way:
- Multiply 1 by each term in [tex]\((3x^2 + 9x + 2)\)[/tex]:
[tex]\[
1 \times 3x^2 = 3x^2
\][/tex]
[tex]\[
1 \times 9x = 9x
\][/tex]
[tex]\[
1 \times 2 = 2
\][/tex]
2. Combine all the products we've obtained:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
After expanding and combining like terms, the final expanded polynomial is:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
This is the result of multiplying [tex]\((x^4 + 1)\)[/tex] by [tex]\((3x^2 + 9x + 2)\)[/tex].
