Question

Write the polynomial in standard form and also write down their degree.
 $ 4p{\text{ }} + {\text{ }}15{p^6} - {\text{ }}{p^5} + {\text{ }}4{p^2}{\text{ }} + {\text{ }}3 $

Answer 1

Hint: A polynomial is in standard form when its terms of highest degree is first then its term of second highest term is second and so on. In this question we first need to determine the degree of terms in expression and then write expression in descending order of degree.

Complete step-by-step answer:
Given polynomial is $ 4p{\text{ }} + {\text{ }}15{p^6} - {\text{ }}{p^5} + {\text{ }}4{p^2}{\text{ }} + {\text{ }}3 $

In eq. 1 the highest exponent is the $ 6 $ , so that term $ 15{p^6} $ is written first in standard form of polynomial.
The next highest exponent is the $ 5 $ so that the term consists of a degree $ 5 $ comes next i.e., $ - {p^5} $ . So far we have $ 15{p^6} - {p^5} $
The next highest exponent is the $ 2 $ so add that term to get
 $ 15{p^6} - {p^5} + 4{p^2} $
Since a variable with no exponent has an understood exponent of 1. Hence it is the next highest exponent, so add that term to get
 $ 15{p^6} - {p^5} + 4{p^2} + 4p $
Finally, the constant term comes because it has no variable i.e. exponent is zero.
So add that term to get final standard form of polynomial is
 $ 15{p^6} - {p^5} + 4{p^2} + 4p + 3 $

Degree is nothing but the highest exponent of the polynomial. So here the highest exponent is 6. So the degree is 6.

Note:Whenever you get this type of question you have to know the meaning of the standard form of polynomial and the procedure that should be followed to get the standard form polynomial. Here are some key points:
Write the term with the highest exponent first
Write the terms with lower exponents in descending order
Remember that a variable with no exponent has an understood exponent of 1
A constant term (a number with no variable) always goes last.