Answer :
To find the correct statement about the sum of the two given polynomials, follow these steps:
1. Write the given polynomials:
[tex]\[ P_1(s, t) = 6s^2t - 2st^2 \][/tex]
[tex]\[ P_2(s, t) = 4s^2t - 3st^2 \][/tex]
2. Sum the two polynomials:
[tex]\[ P(s, t) = P_1(s, t) + P_2(s, t) = (6s^2t - 2st^2) + (4s^2t - 3st^2) \][/tex]
3. Combine like terms:
[tex]\[ P(s, t) = (6s^2t + 4s^2t) + (-2st^2 - 3st^2) \][/tex]
Simplify the coefficients:
[tex]\[ P(s, t) = 10s^2t - 5st^2 \][/tex]
4. Determine the number of terms and the degree:
- The polynomial [tex]\(10s^2t - 5st^2\)[/tex] has two terms.
- To find the degree, identify the exponents of each term:
- In [tex]\(10s^2t\)[/tex], the sum of the exponents is [tex]\(2 (from \ s^2) + 1 (from \ t)\)[/tex], so the degree of this term is [tex]\(2 + 1 = 3\)[/tex].
- In [tex]\(5st^2\)[/tex], the sum of the exponents is [tex]\(1 (from \ s) + 2 (from \ t)\)[/tex], so the degree of this term is [tex]\(1 + 2 = 3\)[/tex].
Both terms have a degree of 3.
Therefore, the polynomial [tex]\(10s^2t - 5st^2\)[/tex] is a binomial with a degree of 3.
Thus, the correct statement is:
The sum is a binomial with a degree of 3.
1. Write the given polynomials:
[tex]\[ P_1(s, t) = 6s^2t - 2st^2 \][/tex]
[tex]\[ P_2(s, t) = 4s^2t - 3st^2 \][/tex]
2. Sum the two polynomials:
[tex]\[ P(s, t) = P_1(s, t) + P_2(s, t) = (6s^2t - 2st^2) + (4s^2t - 3st^2) \][/tex]
3. Combine like terms:
[tex]\[ P(s, t) = (6s^2t + 4s^2t) + (-2st^2 - 3st^2) \][/tex]
Simplify the coefficients:
[tex]\[ P(s, t) = 10s^2t - 5st^2 \][/tex]
4. Determine the number of terms and the degree:
- The polynomial [tex]\(10s^2t - 5st^2\)[/tex] has two terms.
- To find the degree, identify the exponents of each term:
- In [tex]\(10s^2t\)[/tex], the sum of the exponents is [tex]\(2 (from \ s^2) + 1 (from \ t)\)[/tex], so the degree of this term is [tex]\(2 + 1 = 3\)[/tex].
- In [tex]\(5st^2\)[/tex], the sum of the exponents is [tex]\(1 (from \ s) + 2 (from \ t)\)[/tex], so the degree of this term is [tex]\(1 + 2 = 3\)[/tex].
Both terms have a degree of 3.
Therefore, the polynomial [tex]\(10s^2t - 5st^2\)[/tex] is a binomial with a degree of 3.
Thus, the correct statement is:
The sum is a binomial with a degree of 3.
Answer:To determine the correct answer, let's understand how the sum of two polynomials behaves in terms of its degree and number of terms.
Let's denote the two polynomials as \( P(x) \) and \( Q(x) \).
### Understanding Polynomial Addition:
When you add two polynomials \( P(x) \) and \( Q(x) \):
\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \]
\[ Q(x) = b_m x^m + b_{m-1} x^{m-1} + \ldots + b_1 x + b_0 \]
The degree of a polynomial is determined by the highest power of \( x \) with a non-zero coefficient.
### Sum of Polynomials:
When you add \( P(x) \) and \( Q(x) \):
\[ P(x) + Q(x) = (a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0) + (b_m x^m + b_{m-1} x^{m-1} + \ldots + b_1 x + b_0) \]
The resulting polynomial will have terms from both polynomials, and the degree of the resulting polynomial will be the highest degree among the two polynomials \( P(x) \) and \( Q(x) \).
### Answer Analysis:
- **Option A:** The sum is a binomial with a degree of 2.
This would mean both \( P(x) \) and \( Q(x) \) are linear (degree 1) polynomials. However, their sum could result in a polynomial of degree 1 or 2, not necessarily a binomial.
- **Option B:** The sum is a binomial with a degree of 3.
For this to be true, both \( P(x) \) and \( Q(x) \) would need to be quadratic (degree 2) polynomials, but their sum could result in a polynomial of degree 2, 3, or higher, depending on the coefficients.
- **Option C:** The sum is a trinomial with a degree of 2.
This implies \( P(x) \) and \( Q(x) \) are linear polynomials (degree 1), and their sum could indeed result in a polynomial of degree 2, typically represented as \( ax^2 + bx + c \).
- **Option D:** The sum is a trinomial with a degree of 3.
This would mean both \( P(x) \) and \( Q(x) \) are quadratic polynomials (degree 2), and their sum could result in a polynomial of degree 2, 3, or higher.
### Conclusion:
Based on the usual understanding of polynomial addition and the options provided:
- Option A and Option B are unlikely correct because they assume specific degrees without considering the possibility of different degrees.
- Option D is also unlikely correct because it assumes a specific degree without considering other possibilities.
- **Option C** is likely correct because it states that the sum is a trinomial (indicating three terms) with a degree of 2. This allows for the flexibility that the sum could indeed be a quadratic polynomial (degree 2).
Therefore, the correct answer is **C. The sum is a trinomial with a degree of 2.**
Step-by-step explanation:Certainly! Let's go through the reasoning step-by-step to determine why option C is the correct answer.
### Understanding Polynomial Addition:
When we add two polynomials \( P(x) \) and \( Q(x) \), the resulting polynomial \( R(x) = P(x) + Q(x) \) will have terms from both \( P(x) \) and \( Q(x) \). The degree of \( R(x) \), which is the sum of \( P(x) \) and \( Q(x) \), depends on the highest degree term present in either polynomial.
### Given Options:
- **Option A:** The sum is a binomial with a degree of 2.
- **Option B:** The sum is a binomial with a degree of 3.
- **Option C:** The sum is a trinomial with a degree of 2.
- **Option D:** The sum is a trinomial with a degree of 3.
### Analyzing Option C:
**Option C states:** "The sum is a trinomial with a degree of 2."
- **Trinomial:** A trinomial has three terms.
- **Degree of 2:** This refers to the highest power of \( x \) in the polynomial.
### Step-by-Step Explanation:
1. **Degree of Polynomials:**
- If \( P(x) \) is a linear polynomial (degree 1) and \( Q(x) \) is also a linear polynomial (degree 1), their sum \( R(x) = P(x) + Q(x) \) could result in a polynomial of degree 2. This is because the highest degree term in the sum will be \( x^2 \) (if both \( P(x) \) and \( Q(x) \) contribute \( x \cdot x = x^2 \)).
2. **Number of Terms (Trinomial):**
- A trinomial has three terms. If \( P(x) \) and \( Q(x) \) each have a constant term (which is typically included), their sum \( R(x) \) will have three terms.
3. **Degree Analysis:**
- The degree of \( R(x) \), the sum \( P(x) + Q(x) \), will be 2 if \( P(x) \) and \( Q(x) \) are both linear polynomials.
### Conclusion:
Option C correctly states that the sum of the two polynomials is a trinomial (three terms) with a degree of 2 (the highest degree term being \( x^2 \)). This accounts for the scenario where both \( P(x) \) and \( Q(x) \) are linear polynomials, resulting in their sum having a degree of 2.
Therefore, the step-by-step explanation affirms that **option C is the correct answer**: "The sum is a trinomial with a degree of 2."
