Question

State whether the following statement is true or false.
The degree of the sum of two polynomials each of degree $5$ is always $5$.
A. True
B. False

Answer 1

Hint: We will find the answer to this problem with the help of examples considering two polynomials each of degree $5$. We will then add the polynomials to check if the degree of the resulting polynomial is always $5$or not.

Complete step by step Solution :

Let us consider two examples.
$(i)p(x) = 3{x^5} + 1,q(x) = - 3{x^5} + {x^3} + 5$
Here, when we add the two polynomials, we get

$$p(x) + q(x) \\\ = 3{x^5} + 1 + ( - 3{x^5} + {x^3} + 5) \\\ = 3{x^5} + 1 - 3{x^5} + {x^3} + 5 \\\ = {x^3} + 6 \\\$$

We see that the resultant polynomial has a degree of $3$ and not $5$.
Hence, in this example, the degree of the sum of two polynomials each of degree $5$ is not $5$.
$(ii)p(x) = 3{x^5} + 1,q(x) = 3{x^5} + {x^3} + 5$
Here, when we add the two polynomials, we get

$$p(x) + q(x) \\\ = 3{x^5} + 1 + 3{x^5} + {x^3} + 5 \\\ = 6{x^5} + {x^3} + 6 \\\$$

We see that the resultant polynomial has a degree of $5$.
Hence, in this example, the degree of the sum of two polynomials each of degree $5$ is $5$.
Therefore, from the two examples above, we can conclude that the degree of the sum of two polynomials each of degree $5$ is not always $5$.
Thus, the answer is option B.
Note: We see that this question is ambiguous and does give a clear understanding. It gives us both true and false for different examples respectively so we have to go through the problem a few times and have a clear understanding of how to go about solving the problem.