To rewrite the expression [tex]\(8 x^3 y + x^2 - 14 z - 2 + 5 y^2 x\)[/tex] in standard form, we need to arrange the terms in decreasing order of their degree. The degree of each term is determined by the sum of the exponents of the variables within the term.
Here are the degrees of each term in the given expression:
- [tex]\(8 x^3 y\)[/tex] has a degree of 4 (since [tex]\(3 + 1 = 4\)[/tex]),
- [tex]\(5 y^2 x\)[/tex] has a degree of 3 (since [tex]\(2 + 1 = 3\)[/tex]),
- [tex]\(x^2\)[/tex] has a degree of 2,
- [tex]\(-14 z\)[/tex] has a degree of 1,
- [tex]\(-2\)[/tex] has a degree of 0 (as it is a constant).
Now, let's arrange these terms in order of decreasing degree:
1. The highest degree term is [tex]\(8 x^3 y\)[/tex] with degree 4.
2. The next highest degree term is [tex]\(5 y^2 x\)[/tex] with degree 3.
3. Then comes [tex]\(x^2\)[/tex] with degree 2.
4. Next is [tex]\(-14 z\)[/tex] with degree 1.
5. Finally, the constant term [tex]\(-2\)[/tex] with degree 0.
Putting them together, the expression in standard form is:
[tex]\[ 8 x^3 y + 5 y^2 x + x^2 - 14 z - 2 \][/tex]
Therefore, the correct option is:
[tex]\[ 8 x^3 y + 5 y^2 x + x^2 - 14 z - 2 \][/tex]
