Question

How do you write a polynomial in standard form, then classify it by degree and number of term \[{{x}^{4}}{{y}^{2}}+4{{x}^{3}}{{y}^{5}}+10x?\]

Answer 1

Hint: We are given a polynomial as \[{{x}^{4}}{{y}^{2}}+4{{x}^{3}}{{y}^{5}}+10x\] and we have to arrange it into standard form and find the degree and number of terms. To do so we first start by understanding the degree of the polynomial then we will learn how standard equation the polynomial is and then define them. We will arrange the whole polynomial in the required form and then we will find the degree of each term and then arrange it accordingly.

Complete step by step answer:
We are given a polynomial \[{{x}^{4}}{{y}^{2}}+4{{x}^{3}}{{y}^{5}}+10x\] and we have to arrange them in the standard form. To do so we will first observe that the given polynomial is a polynomial composed of 2 variables. Now we have to write it into standard form. Generally, a polynomial is always written in decreasing order of degree of their term, that is, first we will write the term with the highest degree and last we will write the term with the least degree. For example, we have \[{{x}^{2}}+{{x}^{3}}+{{x}^{4}}+x\] then \[{{x}^{4}}\] has the highest degree, then \[{{x}^{3}}\] then \[{{x}^{2}}\] and lastly x. So, in standard form, the equation will be \[{{x}^{4}}+{{x}^{3}}+{{x}^{2}}+x.\]
Now, in our equation, \[{{x}^{4}}{{y}^{2}}+4{{x}^{3}}{{y}^{5}}+10x\] we have two variables. So the degree of each term will be given by the sum of the power of each variable. So, we will find the degree of each term one by one. So, in \[{{x}^{4}}{{y}^{2}}\] we have the power as 4 and 2. So, the degree of \[{{x}^{4}}{{y}^{2}}\] is 4 + 2 = 6.
In \[4{{x}^{3}}{{y}^{5}}\] we have power as 3 and 5, so the degree of \[4{{x}^{3}}{{y}^{5}}\] is 3 + 5 = 8.
In 10x, we have the power as 1 only. So the degree of x is 1.
So, we have the degree as 8, 6 and 1. So, we will arrange the first 8-degree term, then the 6-degree term and then at last 1-degree term. So, we get, \[4{{x}^{3}}{{y}^{5}}+{{x}^{4}}{{y}^{2}}+10x\]

Hence, in the standard form \[{{x}^{4}}{{y}^{2}}+4{{x}^{3}}{{y}^{5}}+10x\] is given as \[4{{x}^{3}}{{y}^{5}}+{{x}^{4}}{{y}^{2}}+10x.\]

Note: Remember in the polynomial of two variables, we need to find the sum of the power to calculate the degree. We can’t say by seeing the highest power. For example, in \[{{x}^{3}}{{y}^{6}}\] and \[{{x}^{1}}{{y}^{7}},\] the first term has the highest power as 6 while the second term has the highest power as 7. So, we cannot say from here that \[{{x}^{1}}{{y}^{7}}\] will come first in standard form. We have to look for the sum and then decide in \[{{x}^{3}}{{y}^{6}}\] the degree is 3 + 6 = 9, while in \[{{x}^{1}}{{y}^{7}}\] the degree is 1 + 7 = 8. So, \[{{x}^{3}}{{y}^{6}}\] has the highest degree. So, it will come ahead of \[x{{y}^{7}}.\]