High School

Multiply:

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

Options:
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 solve the multiplication of the polynomials [tex]\((x^4 + 1)(3x^2 + 9x + 2)\)[/tex], we'll use the distributive property (also known as the FOIL method for binomials) to expand the expression. Here's a step-by-step breakdown:

1. Distribute [tex]\(x^4\)[/tex] across the second polynomial:
- Multiply [tex]\(x^4\)[/tex] by [tex]\(3x^2\)[/tex]:
[tex]\[
x^4 \times 3x^2 = 3x^6
\][/tex]
- Multiply [tex]\(x^4\)[/tex] by [tex]\(9x\)[/tex]:
[tex]\[
x^4 \times 9x = 9x^5
\][/tex]
- Multiply [tex]\(x^4\)[/tex] by [tex]\(2\)[/tex]:
[tex]\[
x^4 \times 2 = 2x^4
\][/tex]

2. Distribute [tex]\(1\)[/tex] across the second polynomial:
- Multiply [tex]\(1\)[/tex] by [tex]\(3x^2\)[/tex]:
[tex]\[
1 \times 3x^2 = 3x^2
\][/tex]
- Multiply [tex]\(1\)[/tex] by [tex]\(9x\)[/tex]:
[tex]\[
1 \times 9x = 9x
\][/tex]
- Multiply [tex]\(1\)[/tex] by [tex]\(2\)[/tex]:
[tex]\[
1 \times 2 = 2
\][/tex]

3. Combine all the terms from the distributions:
- After distributing and multiplying, we gather all terms:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]

So, the expanded form of the expression [tex]\((x^4 + 1)(3x^2 + 9x + 2)\)[/tex] is:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]

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