Answer:
(See explanation for further details)
Step-by-step explanation:
a) Let consider the polynomial . The polynomial is in standards when has the form
, where
is the order of the polynomial. The example has the following information:
,
,
,
,
.
b) The closure property means that polynomials must be closed with respect to addition and multiplication, which is demonstrated hereafter:
Closure with respect to addition:
Let consider polynomials and
such that:
and
, where
Hence, polynomials are closed with respect to addition.
Closure with respect to multiplication:
Let be a polynomial such that:
And an scalar. If the polynomial is multiplied by the scalar number, then:
Lastly, the following expression is constructed by distributive property:
Hence, polynomials are closed with respect to multiplication.