Answer:
a. 6x^5- 3x^3+ 21
b. (6x^5+ 4x^4+21) - (4x^5+ 3x^4+11)= 2x^5+ x^4+10
Explanation:
The degree of the polynomial is the highest power of the terms of the polynomial .e.g
x³+ 2ax+ b² is a polynomial of 3rd degree because the highest power of the terms given is 3.
As we need a polynomial of 5th degree the highest power will be
5.6x^5+ 4x^4+ 3x^3+ 5x^2+ x+21
But this polynomial has 5 terms. We need a 5th degree polynomial with three terms so it can be any one of the following
6x^5- 3x^3+ 21
6x^5+ 4x^4+21
6x^5+ 5x^2+21
6x^5 + x+21
All of the above polynomials are in the standard form because the powers of the terms are given in decreasing order. The first term has the highest power and so on.
Part B:
Suppose we have 2 polynomials
6x^5+ 4x^4+21 and 4x^5+ 3x^4+11
When the closure property of subtraction is applied
(6x^5+ 4x^4+21) - (4x^5+ 3x^4+11)
= 6x^5+ 4x^4+21 - 4x^5- 3x^4-11 → coefficients are changed b/e of subtraction
= 2x^5+ x^4+10
the coefficients are changed because of subtraction.
In Polynomials the sign is changed due to subtraction and same degree terms ( like terms) are subtracted to get the answer.