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.