Answer :
To solve the multiplication problem [tex]\((x^4 + 1)(3x^2 + 9x + 2)\)[/tex], we will expand the expression by distributing each term from the first polynomial across each term of the second polynomial.
Here’s how we do it step-by-step:
1. Distribute [tex]\(x^4\)[/tex] across [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]
So from distributing [tex]\(x^4\)[/tex], we get: [tex]\(3x^6 + 9x^5 + 2x^4\)[/tex]
2. Distribute [tex]\(1\)[/tex] across [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]
So from distributing [tex]\(1\)[/tex], we get: [tex]\(3x^2 + 9x + 2\)[/tex]
3. Combine all the terms:
Now, we add together all these parts from above:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
This is the expanded form of [tex]\((x^4 + 1)(3x^2 + 9x + 2)\)[/tex].
Therefore, the result is [tex]\(3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2\)[/tex]. This represents the fully expanded product of the given polynomials.
Here’s how we do it step-by-step:
1. Distribute [tex]\(x^4\)[/tex] across [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]
So from distributing [tex]\(x^4\)[/tex], we get: [tex]\(3x^6 + 9x^5 + 2x^4\)[/tex]
2. Distribute [tex]\(1\)[/tex] across [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]
So from distributing [tex]\(1\)[/tex], we get: [tex]\(3x^2 + 9x + 2\)[/tex]
3. Combine all the terms:
Now, we add together all these parts from above:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
This is the expanded form of [tex]\((x^4 + 1)(3x^2 + 9x + 2)\)[/tex].
Therefore, the result is [tex]\(3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2\)[/tex]. This represents the fully expanded product of the given polynomials.
