What is true about the sum of the two polynomials? 6s^(2)t-2st^(2) 4s^(2)t-3st^(2) The sum is a binomial with a degree of 2. The sum is a binomial with a degree of 3. The sum is a trinomial with a degree of 2. The sum is a trinomial with a degree of 3.
The correct answer and explanation is :
Let’s solve the problem carefully first:
You are asked:
What is true about the sum of these two polynomials?
Given polynomials:
- ( 6s^2t – 2st^2 )
- ( 4s^2t – 3st^2 )
First, add the polynomials together by combining like terms:
- Combine ( 6s^2t ) and ( 4s^2t ):
[
6s^2t + 4s^2t = 10s^2t
] - Combine ( -2st^2 ) and ( -3st^2 ):
[
-2st^2 + (-3st^2) = -5st^2
]
Thus, the sum is:
[
10s^2t – 5st^2
]
Now, let’s answer the multiple-choice question.
Step 1: Identify if it’s a binomial or trinomial.
- Binomial means two terms.
- Trinomial means three terms.
- In our sum, ( 10s^2t ) and ( -5st^2 ) are two terms → Binomial.
Step 2: Find the degree.
- The degree of a term is the sum of the exponents of all variables in that term.
- For ( 10s^2t ), degree = ( 2 + 1 = 3 ).
- For ( -5st^2 ), degree = ( 1 + 2 = 3 ).
The degree of a polynomial is the highest degree among its terms, so the degree is 3.
Final Answer:
✅ The sum is a binomial with a degree of 3.
300-Word Explanation:
When adding polynomials, you combine like terms—terms that have exactly the same variables raised to the same powers. In this case, the polynomials are ( 6s^2t – 2st^2 ) and ( 4s^2t – 3st^2 ).
- First, combine the terms with ( s^2t ): ( 6s^2t + 4s^2t = 10s^2t ).
- Next, combine the terms with ( st^2 ): ( -2st^2 + (-3st^2) = -5st^2 ).
Adding the polynomials results in:
[
10s^2t – 5st^2
]
This new expression has exactly two terms: ( 10s^2t ) and ( -5st^2 ), meaning it is a binomial (because “bi-” means two).
Next, determine the degree. The degree of a polynomial is based on the term with the highest total degree. For each term:
- ( 10s^2t ) has ( s ) raised to 2 and ( t ) raised to 1. Adding the exponents: ( 2 + 1 = 3 ).
- ( -5st^2 ) has ( s ) raised to 1 and ( t ) raised to 2. Adding the exponents: ( 1 + 2 = 3 ).
Both terms have a degree of 3. Therefore, the degree of the entire binomial is 3.
In conclusion, the sum is not a trinomial (three terms) nor a binomial with degree 2. The correct description is: The sum is a binomial with a degree of 3.