College

Multiply:

[tex](x^4 + 1)(3x^2 + 9x + 2)[/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], follow these steps:

1. Distribute Each Term in the First Polynomial:
- Multiply each term in the first polynomial [tex]\( (x^4 + 1) \)[/tex] by each term in the second polynomial [tex]\( (3x^2 + 9x + 2) \)[/tex].

2. Multiply [tex]\( x^4 \)[/tex] by Each Term in the Second Polynomial:
- [tex]\( x^4 \cdot 3x^2 = 3x^{4+2} = 3x^6 \)[/tex]
- [tex]\( x^4 \cdot 9x = 9x^{4+1} = 9x^5 \)[/tex]
- [tex]\( x^4 \cdot 2 = 2x^4 \)[/tex]

3. Multiply [tex]\( 1 \)[/tex] by Each Term in the Second Polynomial:
- [tex]\( 1 \cdot 3x^2 = 3x^2 \)[/tex]
- [tex]\( 1 \cdot 9x = 9x \)[/tex]
- [tex]\( 1 \cdot 2 = 2 \)[/tex]

4. Combine All These Products:
- Combine all the products obtained:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]

There are no like terms to combine in the polynomial [tex]\( 3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2 \)[/tex], so this is the expanded and final form of the product.