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 found in either of the original polynomials. He gave
wing example

agree with Marcellus? If not, give you counterexample to support your answer.

Ask by Parry Burgess. in the United States

Jan 23,2025

Question

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 found in either of the original polynomials. He gave
wing example

agree with Marcellus? If not, give you counterexample to support your answer.

Ask by Parry Burgess. in the United States

Jan 23,2025

UpStudy AI · Accepted 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.
Marcellus's statement is incorrect. Adding two polynomials does not always result in a polynomial with the highest degree of the original ones. For example, adding $$ x^2 + 2x + 3 $$ and $$ -x^2 + x + 4 $$ gives $$ 3x + 7 $$, which is a first-degree polynomial despite the original polynomials being second-degree.

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 found in either of the original polynomials. He gave
wing example

agree with Marcellus? If not, give you counterexample to support your answer.

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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 found in either of the original polynomials. He gave wing example agree with Marcellus? If not, give you counterexample to support your answer.

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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 found in either of the original polynomials. He gave
wing example

agree with Marcellus? If not, give you counterexample to support your answer.