Certainly! Let's write the given polynomials in standard form. The standard form of a polynomial arranges the terms in descending order of their powers.
### (i) Given Polynomial: [tex]\(x^6 - 3x^4 + \sqrt{2}x + 5x^2 - 2x^5 + 4\)[/tex]
1. Identify and rearrange the terms according to the power of [tex]\(x\)[/tex]:
- [tex]\(x^6\)[/tex]
- [tex]\(-2x^5\)[/tex]
- [tex]\(-3x^4\)[/tex]
- [tex]\(5x^2\)[/tex]
- [tex]\(\sqrt{2}x\)[/tex]
- [tex]\(4\)[/tex]
2. Combine the terms in the correct order:
[tex]\[
x^6 - 2x^5 - 3x^4 + 5x^2 + \sqrt{2}x + 4
\][/tex]
### (ii) Given Polynomial: [tex]\(x^7 - 3x^5 + \sqrt{2}x + \frac{4}{3}x^2 - 2x^6 + 4\)[/tex]
1. Identify and rearrange the terms according to the power of [tex]\(x\)[/tex]:
- [tex]\(x^7\)[/tex]
- [tex]\(-2x^6\)[/tex]
- [tex]\(-3x^5\)[/tex]
- [tex]\(\frac{4}{3}x^2\)[/tex]
- [tex]\(\sqrt{2}x\)[/tex]
- [tex]\(4\)[/tex]
2. Combine the terms in the correct order:
[tex]\[
x^7 - 2x^6 - 3x^5 + \frac{4}{3}x^2 + \sqrt{2}x + 4
\][/tex]
### (iii) Given Polynomial: [tex]\(2x^3 + 3 + x^2 - 3x^5 - x\)[/tex]
1. Identify and rearrange the terms according to the power of [tex]\(x\)[/tex]:
- [tex]\(-3x^5\)[/tex]
- [tex]\(2x^3\)[/tex]
- [tex]\(x^2\)[/tex]
- [tex]\(-x\)[/tex]
- [tex]\(3\)[/tex]
2. Combine the terms in the correct order:
[tex]\[
-3x^5 + 2x^3 + x^2 - x + 3
\][/tex]
### (iv) Given Polynomial: [tex]\(1 + x^3 - 2x^2 - 7x^5\)[/tex]
1. Identify and rearrange the terms according to the power of [tex]\(x\)[/tex]:
- [tex]\(-7x^5\)[/tex]
- [tex]\(x^3\)[/tex]
- [tex]\(-2x^2\)[/tex]
- [tex]\(1\)[/tex]
2. Combine the terms in the correct order:
[tex]\[
-7x^5 + x^3 - 2x^2 + 1
\][/tex]
So the polynomials in standard form are:
(i) [tex]\(x^6 - 2x^5 - 3x^4 + 5x^2 + \sqrt{2}x + 4\)[/tex]
(ii) [tex]\(x^7 - 2x^6 - 3x^5 + \frac{4}{3}x^2 + \sqrt{2}x + 4\)[/tex]
(iii) [tex]\(-3x^5 + 2x^3 + x^2 - x + 3\)[/tex]
(iv) [tex]\(-7x^5 + x^3 - 2x^2 + 1\)[/tex]
