Question

Say 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: Use an example of a polynomial of degree 5 to check if it violates the statement. If it satisfies the statement then it is true otherwise false.

Complete step-by-step answer:
Given, the degree of the sum of two polynomials each of degree 5 is always 5
Let us consider one of the polynomials be equal to \[A(x) = a{x^5} + b{x^4} + c{x^3} + d{x^2} + ex + f\]
And the other polynomial be equal to \[B(x) = - a{x^5} - b{x^4} - c{x^3} - d{x^2} - ex - f\]
Now, we can see that both the polynomials are of degree 5
Hence, consider their sum which is equal to
\[A(x) + B(x) = a{x^5} + b{x^4} + c{x^3} + d{x^2} + ex - a{x^5} - b{x^4} - c{x^3} - d{x^2} - ex - f\]
\[
   = (a - a){x^5} + (b - b){x^4} + (c - c){x^3} + (d - d){x^2} + (e - e)x + (f - f) \\
   = (0){x^5} + (0){x^4} + (0){x^3} + (0){x^2} + (0)x + (0) \\
   = 0 \\
\]
Which is a polynomial of degree 0
Therefore, it contradicts our statement.
Hence, the statement is false. i.e. option (B) is correct.

Note: Only one example is enough to discard a given statement. The degree of a polynomial is the highest power of the variable in a polynomial expression.