Answer :
Sure! Let's multiply the given polynomials step by step: [tex]\((x^4 + 1)\)[/tex] and [tex]\((3x^2 + 9x + 2)\)[/tex].
### Step 1: Distribute each term from [tex]\((x^4 + 1)\)[/tex] with each term from [tex]\((3x^2 + 9x + 2)\)[/tex].
#### Distributing [tex]\(x^4\)[/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]
#### Distributing [tex]\(1\)[/tex]:
- [tex]\(1 \times 3x^2 = 3x^{2}\)[/tex]
- [tex]\(1 \times 9x = 9x\)[/tex]
- [tex]\(1 \times 2 = 2\)[/tex]
### Step 2: Combine all the distributed terms:
The terms we have from the distribution step are:
- [tex]\(3x^{6}\)[/tex]
- [tex]\(9x^{5}\)[/tex]
- [tex]\(2x^{4}\)[/tex]
- [tex]\(3x^{2}\)[/tex]
- [tex]\(9x\)[/tex]
- [tex]\(2\)[/tex]
### Step 3: Write the resulting polynomial:
Combine all the terms to get the final expression:
[tex]\[ 3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2 \][/tex]
This is your multiplied polynomial!
Summary: When you multiply [tex]\((x^4 + 1)\)[/tex] by [tex]\((3x^2 + 9x + 2)\)[/tex], you get the polynomial [tex]\(3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2\)[/tex].
### Step 1: Distribute each term from [tex]\((x^4 + 1)\)[/tex] with each term from [tex]\((3x^2 + 9x + 2)\)[/tex].
#### Distributing [tex]\(x^4\)[/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]
#### Distributing [tex]\(1\)[/tex]:
- [tex]\(1 \times 3x^2 = 3x^{2}\)[/tex]
- [tex]\(1 \times 9x = 9x\)[/tex]
- [tex]\(1 \times 2 = 2\)[/tex]
### Step 2: Combine all the distributed terms:
The terms we have from the distribution step are:
- [tex]\(3x^{6}\)[/tex]
- [tex]\(9x^{5}\)[/tex]
- [tex]\(2x^{4}\)[/tex]
- [tex]\(3x^{2}\)[/tex]
- [tex]\(9x\)[/tex]
- [tex]\(2\)[/tex]
### Step 3: Write the resulting polynomial:
Combine all the terms to get the final expression:
[tex]\[ 3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2 \][/tex]
This is your multiplied polynomial!
Summary: When you multiply [tex]\((x^4 + 1)\)[/tex] by [tex]\((3x^2 + 9x + 2)\)[/tex], you get the polynomial [tex]\(3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2\)[/tex].
