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Answer :
To multiply the polynomials [tex]\((x^4 + 1)(3x^2 + 9x + 2)\)[/tex], we will use the distributive property, also known as the FOIL method for binomials, applying it to each term in the first polynomial with every term in the second polynomial. Here's a step-by-step breakdown of the process:
1. Start with the first term in the first polynomial, [tex]\(x^4\)[/tex]:
- Multiply [tex]\(x^4\)[/tex] by each 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. Move to the second term in the first polynomial, which is [tex]\(1\)[/tex]:
- Multiply [tex]\(1\)[/tex] by each 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 products to form a single polynomial:
- Combine the terms from step 1 and step 2:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
In conclusion, after multiplying [tex]\((x^4 + 1)(3x^2 + 9x + 2)\)[/tex], the resulting polynomial is:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
This matches the combined product terms and provides us with the final result of the multiplication.
1. Start with the first term in the first polynomial, [tex]\(x^4\)[/tex]:
- Multiply [tex]\(x^4\)[/tex] by each 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. Move to the second term in the first polynomial, which is [tex]\(1\)[/tex]:
- Multiply [tex]\(1\)[/tex] by each 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 products to form a single polynomial:
- Combine the terms from step 1 and step 2:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
In conclusion, after multiplying [tex]\((x^4 + 1)(3x^2 + 9x + 2)\)[/tex], the resulting polynomial is:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
This matches the combined product terms and provides us with the final result of the multiplication.
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