Describe the zero vector and the additive inverse of a vecto

Describe the zero vector and the additive inverse of a vector in the vector space of P₈

Solution

Let P₈ be the collection of all real polynomials whose degree is less than or equal to 8 and P8 is vector space.
The zero vector is the polynomial 0(x) such that p(x) + 0(x) = p(x) and 0(x) + p(x) = p(x). Since polynomial addition is equivalent to adding like coefficients, 0(x) must have every coefficient be zero. Hence, 0(x) = 0 + 0x + 0x 2 + 0x 3 . Equivalently, 0(x) = 0
The additive inverse of a polynomial p(x) is the polynomial p(x) such that p(x) + (p(x)) = 0(x). This happens when the coefficients cancel to zero upon polynomial addition.