91Ó°ÊÓ

In mathematics, particularly in linear algebra, the concept of a polynomial basis is fundamental for understanding polynomial expressions in vector spaces. A polynomial basis is essentially a set of polynomials that can be used to express any polynomial in a given space uniquely. For example, in our exercise, the basis

• \(B = \{x+x^2, x-x^2, 1+x\}\)

consists of three polynomials, and any polynomial in the space spanned by these can be expressed as a linear combination of this set. This lets us translate the polynomial problem into a linear algebra problem, where we can apply concepts like vectors and matrices. Understanding the basis allows us to deconstruct and reconstruct polynomials, a critical skill in studying polynomial functions.

A linear combination involves expressing a polynomial as a sum where each term is a product of a coefficient and a basis polynomial. In this exercise, the polynomial is expressed as the linear combination:
\( p(x) = 3(x+x^2) + 1(x-x^2) + 2(1+x) \)
Each coefficient corresponds to the respective polynomial in the basis. Here:

• \(3\) multiplies \(x+x^2\)
• \(1\) multiplies \(x-x^2\)
• \(2\) multiplies \(1+x\)

Using linear combinations, we can efficiently work with complex polynomials and explore different polynomial transformations through the usage of coordinate vectors.

The coordinate vector in the context of a polynomial basis represents the weights, or coefficients, used in forming the linear combination of the basis polynomials. For the exercise, the coordinate vector
\([3, 1, 2]\)
indicates how each basis polynomial contributes to the total polynomial.

• The first coefficient \(3\) shows the contribution from the polynomial \(x+x^2\).
• The second coefficient \(1\) affects the polynomial \(x-x^2\).
• The third coefficient \(2\) acts on \(1+x\).

Understanding coordinate vectors helps in identifying the contribution of each component to the whole polynomial, thereby making it easier to manipulate and simplify polynomial expressions.

Simplifying a polynomial refers to the process of combining like terms and reducing the polynomial to its simplest form. In our exercise, after expressing the polynomial as a linear combination and distributing the scalars, we have:
\( p(x) = 3x + 3x^2 + x - x^2 + 2 + 2x \)
To simplify, we combine like terms:

• Combine the quadratic terms: \(3x^2 - x^2 = 2x^2\)
• Add the linear terms: \(3x + x + 2x = 6x\)
• The constant term remains \(2\).

The simplified polynomial becomes:
\( p(x) = 2x^2 + 6x + 2 \)
Simplification is crucial for obtaining accurate and reduced forms of polynomials, which are easier to work with and analyze in both mathematical and applied contexts.