Question

Cory writes the polynomial x7 + 3x5 + 3x + 1. Melissa writes the polynomial x7 + 5x + 10. Is there a difference between the degree of the sum and the degree of the difference of the polynomials?

Adding their polynomials together or subtracting one polynomial from the other both result in a polynomial with degree 5.

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.

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 1

When added, Cory and Melissa's polynomials result in a polynomial with a degree of 7. In contrast, the degree of the difference between the two polynomials is 5, due to the cancellation of the x^7 terms. This shows that the operations of addition and subtraction on polynomials can result in different degrees for the resulting polynomial.

Step-by-step explanation:

The question revolves around understanding the operations of addition and subtraction of polynomials, and in particular, how these operations affect the degree of the resulting polynomial. Let's tackle the addition and subtraction of the provided polynomials separately.

Firstly, Cory's polynomial is x7 + 3x5 + 3x + 1, and Melissa's polynomial is x7 + 5x + 10. When these polynomials are added, the like terms are combined:
x7 + x7 = 2x7, 3x5 remains unchanged as there is no corresponding term in the second polynomial, and 3x + 5x = 8x. Therefore, the sum is 2x7 + 3x5 + 8x + 11, which clearly has a degree of 7, since that is the highest power of x present in the resulting polynomial.

For subtraction, we subtract Melissa's polynomial from Cory's, which gives: x7 - x7 = 0, 3x5 remains unchanged, and 3x - 5x = -2x. The 1 and -10 simply become -9 when subtracted. The subtraction yields 3x5 - 2x - 9, with a degree of 5, since the highest power of x now is x5 due to the cancellation of the x7 terms.

Thus, adding the polynomials together results in a polynomial with degree 7, while subtracting one polynomial from the other yields a polynomial with degree 5. This correctly answers the question and clarifies that indeed, the degree of the sum is not the same as the degree of the difference between the two given polynomials.

Answer 2

The last option.

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.

Step-by-step explanation:

Both polynomials are of degree 7.

The 7th degree terms are both x^7.

When you add the polynomials, the x^7 term of one polynomial will be added to the x^7 term of the other polynomial resulting in 2x^7. The sum of the polynomials is also of degree 7.

When you subtract the polynomials, the x^7 term of one polynomial subtracted from the other x^7 term of the other polynomial will eliminate the x^7 terms. The difference of the polynomials will have a 3x^5 or a -3x^5 term resulting in a 5ht degree polynomial.

Answer: The last option.