What is true about the sum of the two polynomials?
[tex]
\begin{array}{l}
6 s^2 t-2 s t^2 \\
4 s^2 t-3 s t^2
\end{array}
[/tex]
A. The sum is a binomial with a degree of 2.
B. The sum is a binomial with a degree of 3.
C. The sum is a trinomial with a degree of 2.
D. The sum is a trinomial with a degree of 3.
Answer (1)
Add the two polynomials: ( 6 s 2 t − 2 s t 2 ) + ( 4 s 2 t − 3 s t 2 ) = 10 s 2 t − 5 s t 2 .
Identify the resulting polynomial as a binomial since it has two terms.
Determine the degree of each term: both 10 s 2 t and − 5 s t 2 have a degree of 3.
Conclude that the sum is a binomial with a degree of 3, so the answer is: T h es u mi s abin o mia lw i t ha d e g reeo f 3 .
Explanation
Understanding the Problem We are given two polynomials: 6 s 2 t − 2 s t 2 and 4 s 2 t − 3 s t 2 . We need to find their sum and determine whether the sum is a binomial or a trinomial, and also find its degree.
Adding the Polynomials First, let's add the two polynomials: ( 6 s 2 t − 2 s t 2 ) + ( 4 s 2 t − 3 s t 2 ) Combine like terms: ( 6 s 2 t + 4 s 2 t ) + ( − 2 s t 2 − 3 s t 2 ) 10 s 2 t − 5 s t 2
Identifying the Type of Polynomial Now, let's analyze the resulting polynomial 10 s 2 t − 5 s t 2 . It has two terms: 10 s 2 t and − 5 s t 2 . Therefore, it is a binomial.
Determining the Degree Next, we need to find the degree of the polynomial. The degree of a term is the sum of the exponents of the variables in that term. For the term 10 s 2 t , the exponents are 2 for s and 1 for t . So, the degree of this term is 2 + 1 = 3 .
For the term − 5 s t 2 , the exponents are 1 for s and 2 for t . So, the degree of this term is 1 + 2 = 3 .
The degree of the polynomial is the highest degree among all its terms, which is 3 in this case.
Conclusion Therefore, the sum is a binomial with a degree of 3.
Examples
Polynomials are used to model various real-world phenomena. For example, engineers use polynomials to design bridges and other structures. The degree of a polynomial can represent the complexity of the model. In economics, polynomials can be used to represent cost and revenue functions, where understanding the degree and number of terms can help in analyzing the behavior of these functions. Understanding polynomial addition helps in combining different models or functions to create a more comprehensive representation of a system.
