Linear Algebra Larson book Example 6 Coordinate Representati

Linear Algebra, Larson book

Example 6 “Coordinate Representation in P3, text p209

P=3x^3 - 2x^2 + 4
The standard basis is S={1, x, x^2, x^3}

QUESTION:
Why is the standard basis NOT S={x^3, x^2, x, 1} ?
(If so, the solution would be [P]s=[3 -2 0 4]t)

Solution

Normally basis is written in ascending powers of x.

But in descending also we can write as this also satisfies linear independence and any vector can be written as a linear combination.