Sure! Let's multiply the given algebraic expressions step by step.
We need to evaluate the product of:
[tex]\[ \left(x^3 + x^2 y + x y^2 + y^3\right) \left(22 x^3 + 6 x + 7\right) \][/tex]
To do this, we will distribute each term in the first polynomial across each term in the second polynomial. We'll start by multiplying [tex]\(x^3\)[/tex] with each of the terms in the second polynomial:
1. [tex]\((x^3)(22x^3) = 22x^6\)[/tex]
2. [tex]\((x^3)(6x) = 6x^4\)[/tex]
3. [tex]\((x^3)(7) = 7x^3\)[/tex]
Next, we multiply [tex]\(x^2y\)[/tex] by each term in the second polynomial:
4. [tex]\((x^2y)(22x^3) = 22x^5y\)[/tex]
5. [tex]\((x^2y)(6x) = 6x^3y\)[/tex]
6. [tex]\((x^2y)(7) = 7x^2y\)[/tex]
Next, we multiply [tex]\(xy^2\)[/tex] by each term in the second polynomial:
7. [tex]\((xy^2)(22x^3) = 22x^4y^2\)[/tex]
8. [tex]\((xy^2)(6x) = 6x^2y^2\)[/tex]
9. [tex]\((xy^2)(7) = 7xy^2\)[/tex]
Finally, we multiply [tex]\(y^3\)[/tex] by each term in the second polynomial:
10. [tex]\((y^3)(22x^3) = 22x^3y^3\)[/tex]
11. [tex]\((y^3)(6x) = 6xy^3\)[/tex]
12. [tex]\((y^3)(7) = 7y^3\)[/tex]
Now, we combine all these product terms together to get our final expression:
[tex]\[ 22x^6 + 6x^4 + 7x^3 + 22x^5y + 6x^3y + 7x^2y + 22x^4y^2 + 6x^2y^2 + 7xy^2 + 22x^3y^3 + 6xy^3 + 7y^3 \][/tex]
So, the expanded form of the product is:
[tex]\[ 22x^6 + 22x^5y + 22x^4y^2 + 22x^3y^3 + 6x^4 + 6x^3y + 6x^2y^2 + 6xy^3 + 7x^3 + 7x^2y + 7xy^2 + 7y^3 \][/tex]
And that's our final result!
