High School

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Multiply:

[tex]\left(x^4+1\right)\left(3x^2+9x+2\right)[/tex]

A. [tex]x^4 + 3x^2 + 9x + 3[/tex]

B. [tex]3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2[/tex]

C. [tex]3x^7 + 9x^6 + 2x^5[/tex]

D. [tex]3x^8 + 9x^4 + 2x^4 + 3x^2 + 9x + 2[/tex]

Answer :

To multiply the polynomials [tex]\((x^4 + 1)\)[/tex] and [tex]\((3x^2 + 9x + 2)\)[/tex], we use the distributive property to expand the expression step-by-step.

1. Distribute [tex]\(x^4\)[/tex] across [tex]\((3x^2 + 9x + 2)\)[/tex]:

- Multiply [tex]\(x^4 \cdot 3x^2 = 3x^{6}\)[/tex].
- Multiply [tex]\(x^4 \cdot 9x = 9x^{5}\)[/tex].
- Multiply [tex]\(x^4 \cdot 2 = 2x^{4}\)[/tex].

So, the expanded form from this distribution is:
[tex]\[
3x^{6} + 9x^{5} + 2x^{4}
\][/tex]

2. Distribute [tex]\(1\)[/tex] across [tex]\((3x^2 + 9x + 2)\)[/tex]:

- Multiply [tex]\(1 \cdot 3x^2 = 3x^2\)[/tex].
- Multiply [tex]\(1 \cdot 9x = 9x\)[/tex].
- Multiply [tex]\(1 \cdot 2 = 2\)[/tex].

So, the expanded form from this distribution is:
[tex]\[
3x^2 + 9x + 2
\][/tex]

3. Combine all the terms:

Combine the results from step 1 and step 2:
[tex]\[
3x^{6} + 9x^{5} + 2x^{4} + 3x^2 + 9x + 2
\][/tex]

This gives us the expanded and combined result for the multiplication of the two polynomials:
[tex]\[
3x^{6} + 9x^{5} + 2x^{4} + 3x^2 + 9x + 2
\][/tex]

And this is your final answer.

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