Question

How do you write the polynomial so that the exponents decrease from left to right, identify the degree and the leading coefficient of the polynomial \[9{m^5}\]?

Answer 1

Hint: Here, we will find the degree of the polynomial from the leading coefficient of the polynomial. We will then write a polynomial starting with the leading coefficient of the polynomial and decreasing the exponent of the variable by 1 until we reach the exponent as 0. A polynomial is a combination of constants and variables along with its coefficient.

Formula Used:
Zero exponent Rule: \[{m^0} = 1\]

Complete Step by Step Solution:
We are given that the leading coefficient of the polynomial \[9{m^5}\].
Now, we will identify the degree of the polynomial from the leading coefficient of the polynomial.
Since the leading coefficient of the polynomial \[9{m^5}\], so the degree of the polynomial is \[5\].
Now, we will write the polynomial so that the exponents decrease from left to right from the leading coefficient of the polynomial.
Let \[f\left( x \right)\] be the polynomial.
\[ \Rightarrow f\left( x \right) = 9{m^5} + {m^4} + {m^3} + {m^2} + {m^1} + {m^0}\]
Zero exponent Rule: \[{m^0} = 1\]
Now, by using the zero exponent rule, we get
\[ \Rightarrow f\left( x \right) = 9{m^5} + {m^4} + {m^3} + {m^2} + {m^1} + 1\]

Therefore, the degree of the polynomial is \[5\] and the polynomial with the exponents decreasing from left to right from the leading coefficient of the polynomial is \[f\left( x \right) = 9{m^5} + {m^4} + {m^3} + {m^2} + {m^1} + 1\].

Note:
We know that the degree of the polynomial is defined as the highest exponent in the given Polynomial function. We know that the leading coefficient of the polynomial is defined as the coefficient corresponding to the highest degree in the given polynomial. A polynomial function is defined as a function which has more than three terms. The standard form of a polynomial is a method where the highest degree of the polynomial is written at first while writing a polynomial. We should remember that the constant term has no variables and it contains any number. So, we have many polynomials with the given leading coefficient. If we add a number of polynomials, we will get a polynomial.