Cory writes the polynomial [tex]x^7+3 x^5+3 x+1[/tex]. Melissa writes the polynomial [tex]x^7+5 x+10[/tex]. Is there a difference between the degree of the sum and the degree of the difference of the polynomials?
A. Adding their polynomials together or subtracting one polynomial from the other both result in a polynomial with degree 7.
B. Adding their polynomials together or subtracting one polynomial from the other both result in a polynomial with degree 5.
C. Adding their polynomials together results in a polynomial with degree 14, but subtracting one polynomial from the other results in a polynomial with degree 5.
D. Adding their polynomials together results in a polynomial with degree 7, but subtracting one polynomial from the other results in a polynomial with degree 5.
Answer (2)
The degrees of the sum and difference of the polynomials are 7 and 5, respectively, indicating a difference. Thus, the correct answer is option D. Cory's and Melissa's polynomials exhibit this discrepancy in their degrees when added and subtracted.
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Add the two polynomials: ( x 7 + 3 x 5 + 3 x + 1 ) + ( x 7 + 5 x + 10 ) = 2 x 7 + 3 x 5 + 8 x + 11 , which has degree 7.
Subtract the two polynomials: ( x 7 + 3 x 5 + 3 x + 1 ) − ( x 7 + 5 x + 10 ) = 3 x 5 − 2 x − 9 , which has degree 5.
Compare the degrees: 7 = 5 .
Conclude that there is a difference between the degree of the sum and the degree of the difference: Yes .
Explanation
Understanding the Problem We are given two polynomials: Cory's polynomial: x 7 + 3 x 5 + 3 x + 1 Melissa's polynomial: x 7 + 5 x + 10 We want to determine if the degree of the sum of the polynomials is different from the degree of the difference of the polynomials.
Finding the Sum First, let's find the sum of the two polynomials: ( x 7 + 3 x 5 + 3 x + 1 ) + ( x 7 + 5 x + 10 ) = ( 1 + 1 ) x 7 + 3 x 5 + ( 3 + 5 ) x + ( 1 + 10 ) = 2 x 7 + 3 x 5 + 8 x + 11 The degree of the sum is 7, since the highest power of x with a non-zero coefficient is 7.
Finding the Difference Next, let's find the difference of the two polynomials (Cory's - Melissa's): ( x 7 + 3 x 5 + 3 x + 1 ) − ( x 7 + 5 x + 10 ) = ( 1 − 1 ) x 7 + 3 x 5 + ( 3 − 5 ) x + ( 1 − 10 ) = 0 x 7 + 3 x 5 − 2 x − 9 = 3 x 5 − 2 x − 9 The degree of the difference is 5, since the highest power of x with a non-zero coefficient is 5.
Comparing the Degrees The degree of the sum is 7, and the degree of the difference is 5. Since 7 = 5 , there is a difference between the degree of the sum and the degree of the difference of the polynomials.
Final Answer Therefore, the answer is yes, there is a difference between the degree of the sum and the degree of the difference of the polynomials.
Examples
Polynomials are used in many areas of mathematics and science. For example, they are used to model curves in computer graphics, to describe the motion of projectiles in physics, and to represent the relationships between variables in economics. Understanding the degree of a polynomial can help us understand the behavior of the function it represents. In this case, we are looking at the degree of the sum and difference of two polynomials. This can be useful in situations where we are combining or comparing different models or relationships.
