To order the polynomial [tex]\(-x^4 y^2 + 7 x^3 y^3 - 3 x y^5 + 2 x^2 y^4\)[/tex] in descending powers of [tex]\(x\)[/tex], let's follow these steps:
1. Identify the power of [tex]\(x\)[/tex] in each term:
- For the term [tex]\(-x^4 y^2\)[/tex], the power of [tex]\(x\)[/tex] is [tex]\(4\)[/tex].
- For the term [tex]\(7 x^3 y^3\)[/tex], the power of [tex]\(x\)[/tex] is [tex]\(3\)[/tex].
- For the term [tex]\(-3 x y^5\)[/tex], the power of [tex]\(x\)[/tex] is [tex]\(1\)[/tex].
- For the term [tex]\(2 x^2 y^4\)[/tex], the power of [tex]\(x\)[/tex] is [tex]\(2\)[/tex].
2. List the terms and their corresponding powers of [tex]\(x\)[/tex]:
- [tex]\(-x^4 y^2\)[/tex] (power of [tex]\(x = 4\)[/tex])
- [tex]\(7 x^3 y^3\)[/tex] (power of [tex]\(x = 3\)[/tex])
- [tex]\(2 x^2 y^4\)[/tex] (power of [tex]\(x = 2\)[/tex])
- [tex]\(-3 x y^5\)[/tex] (power of [tex]\(x = 1\)[/tex])
3. Order the terms based on the descending powers of [tex]\(x\)[/tex]:
- Highest power first: [tex]\(-x^4 y^2\)[/tex] (power [tex]\(4\)[/tex])
- Next highest power: [tex]\(7 x^3 y^3\)[/tex] (power [tex]\(3\)[/tex])
- Followed by: [tex]\(2 x^2 y^4\)[/tex] (power [tex]\(2\)[/tex])
- Lowest power last: [tex]\(-3 x y^5\)[/tex] (power [tex]\(1\)[/tex])
4. Write the polynomial with the terms in the correct order:
[tex]\[
-x^4 y^2 + 7 x^3 y^3 + 2 x^2 y^4 - 3 x y^5
\][/tex]
Thus, the polynomial [tex]\(-x^4 y^2 + 7 x^3 y^3 - 3 x y^5 + 2 x^2 y^4\)[/tex], when ordered in descending powers of [tex]\(x\)[/tex], is:
[tex]\[
-x^4 y^2 + 7 x^3 y^3 + 2 x^2 y^4 - 3 x y^5
\][/tex]
