Question

Two polynomials of distinct degrees m and n are added. The resulting polynomial has a degree: -
(a) Which could be greater than both m and n.
(b) Which must be equal to either m or n.
(c) Which could or could not be equal to either m or n.
(d) Which could be less than both m and n.

Answer 1

Hint: Understand the term ‘polynomial’ and ‘degree of a polynomial’. Form two cases such that, in case (i) assume that m is greater than n and in case (ii) assume that n is greater than m. Take examples in both the cases and add them to determine the correct answer in the given options.

Complete step-by-step solution:
Here, we have been provided with two polynomials of degree m and n and they are added. We have to determine the degree of the resultant polynomial. Let us first see what is a polynomial.
In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves the operations of addition, subtraction, multiplication, and non – negative integer exponentiation of variables. Examples of polynomials can be \[{{x}^{2}}-4x+8,x{{y}^{3}}+y{{z}^{2}}+xz+1\] etc.
Now, let us see what is the degree of the polynomial.
In mathematics, the degree of a polynomial is the highest sum of the exponents of the variables that appear in it. Let us consider the above examples: -
(i) \[{{x}^{2}}-4x+8{{x}^{2}}={{x}^{2}}-4{{x}^{1}}+8{{x}^{0}}\]
Here, the only variable is x and we can see that the highest power of x in all the three terms is 2. Hence, the degree of the polynomial is 2.
(ii) \[x{{y}^{3}}+y{{z}^{2}}+xz+1={{x}^{1}}{{y}^{3}}+{{y}^{1}}{{z}^{2}}+{{x}^{1}}{{z}^{1}}+{{x}^{0}}{{y}^{0}}{{z}^{0}}\]
Here, we can see that there are three variables, i.e., x, y and z. Now, sum of the exponents of variables in \[{{x}^{1}}{{y}^{3}}\] is 4. Similarly, sum of the exponents of variables in \[{{y}^{1}}{{z}^{2}},{{x}^{1}}{{z}^{1}}\] and \[{{x}^{0}}{{y}^{0}}{{z}^{0}}\] are 3, 2 and 0 respectively. Therefore, the highest sum of the exponents is 4 and hence, here the degree of the polynomial is 4.
Now, let us come to the question. We have two polynomials of degree m and n. So, let us assume them as: -
\[\begin{align}
  & \Rightarrow f\left( x \right)={{x}^{m}} \\
 & \Rightarrow g\left( x \right)={{x}^{n}} \\
 & \Rightarrow f\left( x \right)+g\left( x \right)={{x}^{m}}+{{x}^{n}} \\
\end{align}\]
So, we can have the following two cases: -
1. Case (i): - When m > n.
In this case \[f\left( x \right)+g\left( x \right)={{x}^{m}}+{{x}^{n}}\] will have the degree m because the highest power of x in both the term is m. So, degree = m.
2. Case (ii): - When n > m.
In this case \[f\left( x \right)+g\left( x \right)={{x}^{m}}+{{x}^{n}}\] will have the degree n because the highest power of x in both the term is n. So, degree = n.
From the above two cases, we can conclude that the degree of the resulting polynomial will depend on the values of m and n. So, the degree can be either m or n depending on which of these is greater.
Hence, option (b) is the correct answer.

Note: One may note that we have taken simple examples of the two polynomials \[f\left( x \right)\] and \[g\left( x \right)\]. This is because we do not need any complex polynomial example to get the answer. One thing you must note that m and n cannot be negative or any fractional number. They must be some positive integer or 0. This is because polynomials must not contain negative or fractional exponents of any variables.