Answer :
To multiply the polynomials [tex]\( (x^4 + 1) \)[/tex] and [tex]\( (3x^2 + 9x + 2) \)[/tex], we should apply the distributive property, also known as the FOIL method in simpler cases. Here's a step-by-step breakdown of the process:
1. Distribute each term in the first polynomial to every term in the second polynomial.
The first polynomial is [tex]\( x^4 + 1 \)[/tex] and the second polynomial is [tex]\( 3x^2 + 9x + 2 \)[/tex].
2. Multiply [tex]\( x^4 \)[/tex] by each term in the second polynomial:
- [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]
3. Multiply [tex]\( 1 \)[/tex] by each term in the second polynomial:
- [tex]\( 1 \times 3x^2 = 3x^2 \)[/tex]
- [tex]\( 1 \times 9x = 9x \)[/tex]
- [tex]\( 1 \times 2 = 2 \)[/tex]
4. Combine all the products:
Bring together all the terms obtained from the multiplication:
- [tex]\( 3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2 \)[/tex]
5. Simplify the expression:
Since there are no like terms to combine, the final expression remains as:
- [tex]\( 3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2 \)[/tex]
Therefore, the result of multiplying these two polynomials is [tex]\( 3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2 \)[/tex].
1. Distribute each term in the first polynomial to every term in the second polynomial.
The first polynomial is [tex]\( x^4 + 1 \)[/tex] and the second polynomial is [tex]\( 3x^2 + 9x + 2 \)[/tex].
2. Multiply [tex]\( x^4 \)[/tex] by each term in the second polynomial:
- [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]
3. Multiply [tex]\( 1 \)[/tex] by each term in the second polynomial:
- [tex]\( 1 \times 3x^2 = 3x^2 \)[/tex]
- [tex]\( 1 \times 9x = 9x \)[/tex]
- [tex]\( 1 \times 2 = 2 \)[/tex]
4. Combine all the products:
Bring together all the terms obtained from the multiplication:
- [tex]\( 3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2 \)[/tex]
5. Simplify the expression:
Since there are no like terms to combine, the final expression remains as:
- [tex]\( 3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2 \)[/tex]
Therefore, the result of multiplying these two polynomials is [tex]\( 3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2 \)[/tex].
