Critique the reasoning of others. Marcellus made the statement that the sum of two polynomials is al homial with degree equal to the highest power of \( x \) found in either of the original polynomials. He gave wing example \( +5 x+4)+(6 x+1)=3 x^{2}+11 x+5 \) agree with Marcellus? If not, give you counterexample to support your answer.

Marcellus's statement contains a misunderstanding regarding the degrees of polynomials when they are added together. Let's break down his claim and analyze it critically. ### **Marcellus's Claim:** - **Statement:** *The sum of two polynomials is a polynomial with degree equal to the highest power of \( x \) found in either of the original polynomials.* - **Example Provided:** \[ (5x + 4) + (6x + 1) = 3x^{2} + 11x + 5 \] ### **Analysis of the Example:** 1. **Original Polynomials:** - \( P(x) = 5x + 4 \)  **(Degree 1)** - \( Q(x) = 6x + 1 \)  **(Degree 1)** 2. **Sum of Polynomials:** \[ P(x) + Q(x) = (5x + 4) + (6x + 1) = 11x + 5 \] - **Resulting Polynomial:** \( 11x + 5 \)  **(Degree 1)** 3. **Discrepancy Identified:** - Marcellus claims the sum is \( 3x^{2} + 11x + 5 \) (Degree 2). - However, the correct sum is \( 11x + 5 \) (Degree 1). ### **Critique of Marcellus's Reasoning:** - **Incorrect Example:** The example provided by Marcellus is mathematically incorrect. Adding two first-degree polynomials should not yield a second-degree polynomial unless there are higher-degree terms in the original polynomials, which is not the case here. - **General Statement Misconception:** Even if Marcellus's example were correct, his general statement overlooks scenarios where the highest-degree terms cancel each other out upon addition, resulting in a polynomial of lower degree than the maximum of the original degrees. ### **Counterexample to Support the Critique:** Let's consider two polynomials where the highest-degree terms cancel each other out when summed. - **Polynomial 1:** \( P(x) = x^{2} + 2x + 3 \)  **(Degree 2)** - **Polynomial 2:** \( Q(x) = -x^{2} + x + 4 \)  **(Degree 2)** **Sum of Polynomials:** \[ P(x) + Q(x) = (x^{2} + 2x + 3) + (-x^{2} + x + 4) = (x^{2} - x^{2}) + (2x + x) + (3 + 4) = 3x + 7 \] - **Resulting Polynomial:** \( 3x + 7 \)  **(Degree 1)** **Conclusion from Counterexample:** Despite both original polynomials being of degree 2, their sum is a first-degree polynomial. This shows that Marcellus's assertion is not universally valid. ### **Correct Statement:** The **degree of the sum** of two polynomials is **at most** the highest degree among the original polynomials. It **equals** the highest degree **unless** the leading terms (highest-degree terms) cancel each other out during the addition. ### **Final Verdict:** Marcellus's reasoning is **incorrect** both in his provided example and in his general statement. The degree of the sum of two polynomials does not always equal the highest degree of the original polynomials; it can be lower if leading terms cancel each other.