Answer :
To multiply the polynomials [tex]\((x^4 + 1)\)[/tex] and [tex]\((3x^2 + 9x + 2)\)[/tex], we can use the distributive property to expand the expression step by step.
Let's break down the multiplication process:
1. Multiply each term in the first polynomial by each term in the second polynomial:
- [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]
- [tex]\(1 \cdot 3x^2 = 3x^2\)[/tex]
- [tex]\(1 \cdot 9x = 9x\)[/tex]
- [tex]\(1 \cdot 2 = 2\)[/tex]
2. Add all the resulting terms together:
Combine all the results from above:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
Since none of the resulting terms are like terms (none have the same variable raised to the same power), you just list them together.
Therefore, the expanded form of the product [tex]\((x^4 + 1)(3x^2 + 9x + 2)\)[/tex] is:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
This is the simplified expression you get after multiplying the two polynomials.
Let's break down the multiplication process:
1. Multiply each term in the first polynomial by each term in the second polynomial:
- [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]
- [tex]\(1 \cdot 3x^2 = 3x^2\)[/tex]
- [tex]\(1 \cdot 9x = 9x\)[/tex]
- [tex]\(1 \cdot 2 = 2\)[/tex]
2. Add all the resulting terms together:
Combine all the results from above:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
Since none of the resulting terms are like terms (none have the same variable raised to the same power), you just list them together.
Therefore, the expanded form of the product [tex]\((x^4 + 1)(3x^2 + 9x + 2)\)[/tex] is:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
This is the simplified expression you get after multiplying the two polynomials.
