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.
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.
