Question

In case of a polynomial in one variable, the highest power of the variable is called the degree of the polynomial.
In case of polynomials in more than one variable, the sum of the powers of the variables in each term is taken up and the highest sum obtained is called the degree of the polynomial.
Find the polynomial with degree 6.
(a) $6{{x}^{4}}+7{{x}^{2}}$
(b) $8{{x}^{4}}{{y}^{2}}+7{{x}^{5}}{{y}^{3}}-\dfrac{3}{5}$
(c) $8{{x}^{2}}{{y}^{2}}{{z}^{2}}+7{{x}^{2}}{{y}^{3}}z+3{{x}^{4}}$
(d) $8{{x}^{6}}{{y}^{6}}+{{y}^{2}}+7{{x}^{3}}$

Answer 1

Hint: Check each option by taking the sum of the power of the variables in each term. If the highest sum of the powers of the variables of a term among all the terms in a particular option is 6, then that option is the required answer.

Complete step-by-step solution -
Let us check in option (a).
The given polynomial is $6{{x}^{4}}+7{{x}^{2}}$. It contains two terms, which are: $6{{x}^{4}}$ and $7{{x}^{2}}$.
Sum of powers of the variable in $6{{x}^{4}}$ = 4
Sum of powers of the variable in $7{{x}^{2}}$ = 2
Therefore, the highest sum of the powers of the variable is 4.

Let us check in option (b).
The given polynomial is $8{{x}^{4}}{{y}^{2}}+7{{x}^{5}}{{y}^{3}}-\dfrac{3}{5}$. It contains three terms, which are: $8{{x}^{4}}{{y}^{2}}$, $7{{x}^{5}}{{y}^{3}}$ and $-\dfrac{3}{5}$.
Sum of powers of the variable in $8{{x}^{4}}{{y}^{2}}$ = 4 + 2 = 6
Sum of powers of the variable in $7{{x}^{5}}{{y}^{3}}$ = 5 + 3 = 8
Sum of powers of the variable in $-\dfrac{3}{5}$ = 0
Therefore, the highest sum of the powers of the variable is 8.

Let us check in option (c).
The given polynomial is $8{{x}^{2}}{{y}^{2}}{{z}^{2}}+7{{x}^{2}}{{y}^{3}}z+3{{x}^{4}}$. It contains three terms, which are: $8{{x}^{2}}{{y}^{2}}{{z}^{2}}$, $7{{x}^{2}}{{y}^{3}}z$ and $3{{x}^{4}}$.
Sum of powers of the variable in $8{{x}^{2}}{{y}^{2}}{{z}^{2}}$ = 2 + 2 + 2 = 6
Sum of powers of the variable in $7{{x}^{2}}{{y}^{3}}z$ = 2 + 3 + 1 = 6
Sum of powers of the variable in $3{{x}^{4}}$ = 4
Therefore, the highest sum of the powers of the variable is 6.

Let us check in option (d).
The given polynomial is $8{{x}^{6}}{{y}^{6}}+{{y}^{2}}+7{{x}^{3}}$. It contains three terms, which are: $8{{x}^{6}}{{y}^{6}}$, ${{y}^{2}}$ and $7{{x}^{3}}$.
Sum of powers of the variable in $8{{x}^{6}}{{y}^{6}}$ = 6 + 6 = 12
Sum of powers of the variable in ${{y}^{2}}$ = 2
Sum of powers of the variable in $7{{x}^{3}}$ = 3
Therefore, the highest sum of the powers of the variable is 12.

So, on checking all the terms of all the options, we get that, highest sum of the powers of the variable is 6 in the polynomial $8{{x}^{2}}{{y}^{2}}{{z}^{2}}+7{{x}^{2}}{{y}^{3}}z+3{{x}^{4}}$.
Hence, option (c) is the correct answer.

Note: It is important to check all the terms carefully one by one to get the answer. As we can see that the option (b) contains a term whose sum of power of variables is 6 but it also contains a term whose sum of power of variables is 8. Therefore, the highest power is 8, so it is a polynomial of degree 8. So, don’t get confused about such things.