Understanding the Vector Space Properties of Polynomials of Degree Two

When considering the set of polynomials of degree exactly two, it is essential to determine if this set satisfies the axioms required for a vector space. This article will explore the key properties, particularly the closure under addition and scalar multiplication, to understand why the set of polynomials of degree two does not form a vector space.

Introduction to Polynomials of Degree Two

Let's denote the set of polynomials of degree exactly two as ( P_2 ). A general form of such a polynomial can be written as:

[ P(x) ax^2 bx c ]

where ( a eq 0 ) to ensure the polynomial is of degree two.

Closure Under Addition

To check for closure under addition, we need to verify that if we take any two polynomials from ( P_2 ), their sum also belongs to ( P_2 ). Consider two polynomials from ( P_2 ): [ P_1(x) a_1x^2 b_1x c_1 ] [ P_2(x) a_2x^2 b_2x c_2 ] When we add these two polynomials, we get:

[ P_1(x) P_2(x) (a_1x^2 b_1x c_1) (a_2x^2 b_2x c_2) (a_1 a_2)x^2 (b_1 b_2)x (c_1 c_2) ]

The resulting polynomial is still a polynomial of degree two if both ( a_1 ) and ( a_2 ) are non-zero. However, if either ( a_1 ) or ( a_2 ) are zero, the resulting polynomial may have a lower degree. For instance, if ( a_1 0 ) and ( a_2 eq 0 ), the sum may have a degree less than two.

Example

Consider the polynomials:

[ P_1(x) 3x^2 4x 5 ] [ P_2(x) ^2 6x 2 ]

When added, the polynomial becomes:

[ P_1(x) P_2(x) (3x^2 4x 5) (^2 6x 2) 3x^2 1 7 ]

This is still a polynomial of degree two. However, if we consider:

[ P_1(x) 3x^2 4x 5 ] [ P_2(x) -3x^2 6x 2 ]

When added, the polynomial becomes:

[ P_1(x) P_2(x) (3x^2 4x 5) (-3x^2 6x 2) 1 7 ]

The resulting polynomial is a polynomial of degree one, thus not satisfying closure under addition.

Closure Under Scalar Multiplication

To check for closure under scalar multiplication, we need to verify if multiplying any polynomial from ( P_2 ) by a scalar results in another polynomial from ( P_2 ). Consider a polynomial from ( P_2 ) and a scalar ( k ): [ P(x) ax^2 bx c ] When multiplied by a scalar ( k ), we get:

[ k cdot P(x) k(ax^2 bx c) kax^2 kbx kc ]

For the product to remain in ( P_2 ), the coefficient ( ka ) must be non-zero. If ( k 0 ), the product is:

[ 0 cdot (ax^2 bx c) 0 ]

This results in the zero polynomial, which is a polynomial of degree less than two, hence not in ( P_2 ).

Conclusion

Due to the failure of closure under addition and scalar multiplication, the set of polynomials of degree exactly two does not form a vector space. Therefore, the set of polynomials of degree exactly two does not satisfy the axioms of a vector space.

Summary

1. Failure of closure under addition: The sum of two polynomials from ( P_2 ) may result in a polynomial of lower degree if one of the polynomials has a zero coefficient for the ( x^2 ) term.

2. Failure of closure under scalar multiplication: Multiplying a polynomial from ( P_2 ) by zero results in the zero polynomial, which is not a polynomial of degree two.

Thus, the set of polynomials of degree two does not form a vector space.