High School

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 solve the multiplication of the polynomials [tex]\((x^4 + 1)\)[/tex] and [tex]\((3x^2 + 9x + 2)\)[/tex], we will expand the expression step-by-step using the distributive property.

1. Expand each term of [tex]\( x^4 + 1 \)[/tex] with the terms in [tex]\( 3x^2 + 9x + 2 \)[/tex]:

- Start with [tex]\( x^4 \)[/tex]:

- Multiply [tex]\( x^4 \)[/tex] by each term in [tex]\((3x^2 + 9x + 2)\)[/tex]:
[tex]\[
x^4 \times 3x^2 = 3x^{6}
\][/tex]
[tex]\[
x^4 \times 9x = 9x^5
\][/tex]
[tex]\[
x^4 \times 2 = 2x^4
\][/tex]

- Now expand 1 in the same way:

- Multiply 1 by each term in [tex]\((3x^2 + 9x + 2)\)[/tex]:
[tex]\[
1 \times 3x^2 = 3x^2
\][/tex]
[tex]\[
1 \times 9x = 9x
\][/tex]
[tex]\[
1 \times 2 = 2
\][/tex]

2. Combine all the products we've obtained:

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

After expanding and combining like terms, the final expanded polynomial is:

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

This is the result of multiplying [tex]\((x^4 + 1)\)[/tex] by [tex]\((3x^2 + 9x + 2)\)[/tex].