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8.22. Let S be the set of all polynomials of degree at most 3. An element s(x) of S can then be expressed as
s(x) = ax3 + bx2 + cx + d, where a, b, c, d ? R. A relation R is defined on S by p(x) R q(x) if p(x) and
q(x) have a real root in common. (For example, p = (x ? 1)^2 and q = x^2 ? 1 have the root 1 in common so
that p R q.) Determine which of the properties reflexive, symmetric, and transitive are possessed by R.

Answer :

R possess Symmetric property.

Let S be the set of all polynomials of degree at most 3.

s(x) = ax³ + bx² + cx + d

we know that polynomial is a mathematical expression of one or more algebraic terms each of which consist of a constant multiplied by one or more variables raised to a non negative power.  

The reflexive property states that every element of the set is related to itself.

The symmetric property is defined as if one element in a set is related to the other, then we can say that the second element is also related to the first element.

The transitive property states that if a, b, c are three quantities, and if a is related to b and b is related to c then a and c are related  to each other.

So here we observe that R will posses symmetric property.

To know more about property here

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