Question

Which one of the following polynomials is in standard form?
A. $$5{x^7} - {x^1} + 2{x^4} - {x^2}$$
B. $$9{x^8} - 4{x^2} + 1 - {x^5}$$
C. $$7{x^3} + 9{x^2} - {x^2} + 1 - {x^9}$$
D. $${x^6} - {x^4} - {x^2} + 1$$

Answer 1

Hint: Here in this question given the list of polynomials, we have to identify the polynomial which is in standard form. For this need to observe or focus on the degree of the terms rather than the coefficient in standard form of polynomial the term with highest power or degree comes first and the degree is in descending order from first to last.

Complete answer:
Variables, coefficients and constants, that involves only the operations of addition, subtraction, multiplication,
and non-negative integer exponentiation of variables. The degree of a polynomial is the highest power of the variable in the polynomial.
The standard form of a polynomial represented as:
$${a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + \cdots + {a_2}{x^2} + {a_1}{x^1} + {a_0}{x^0}$$ Standard form of a polynomial should write the polynomial as the sum of the decreasing power of the variable '$$x$$'. 
Now, consider the given question:
We have to identify the polynomial which is in standard form by giving the following polynomials.
A. $$5{x^7} - {x^1} + 2{x^4} - {x^2}$$
Here, powers are not arranged in descending order. So, it’s not in standard form.

B. $$9{x^8} - 4{x^2} + 1 - {x^5}$$
Here, powers are not arranged in descending order. So, it’s not in standard form.

C. $$7{x^3} + 9{x^2} - {x^2} + 1 - {x^9}$$
Here, powers are not arranged in descending order. So, it’s not in standard form.
D. $${x^6} - {x^4} - {x^2} + 1$$
Here, powers are arranged in descending order, meaning the term with the highest degree or power first, then the term with the next highest degree, and so on.
So, is $${x^6} - {x^4} - {x^2} + 1$$ a standard form of the polynomial.$$$$

Therefore, the correct option is D

Note: A polynomial can have constants, variables and exponents but never negative exponents or division by a variable and zeros should be considered as the roots for getting the required factors by assuming the polynomial is equal to '0'. Polynomials like to be written in standard form means they contain no like terms and the exponents are in descending order i.e., largest to smallest.