Final answer:
The student's question involves adding two polynomials with different exponents, resulting in another polynomial. This process demonstrates that the set of polynomials is closed under the operation of addition, as the sum of any two polynomials is still a polynomial.
Step-by-step explanation:
The student is asking to demonstrate that the set of polynomials is closed under the operation of addition. To show this, we add the polynomials 3x4 + 6x-7 and 2x3 - (2x)2 + 4x-1. It's important to combine like terms, which are terms that have the same variables raised to the same powers. However, here there are no like terms since all the terms in both polynomials have different exponents.
The addition of these two polynomials simply involves writing them together, because there are no common terms to combine: 3x4 + 2x3 - 4x2 + 4x-1 + 6x-7. The result is indeed a polynomial, which verifies that polynomials are closed under addition.
To add the polynomials 3x⁴+6x⁻⁷ and 2x³-(2x)²+4x⁻¹, we first group the like terms. The like terms have the same variable raised to the same exponent. In this case, the like terms are:
3x⁴ and 2x³
6x⁻⁷ and 4x⁻¹
(2x)²
Next, we add the coefficients of the like terms. So, the resulting polynomial is: 3x⁴ + 2x³ + 6x⁻⁷ + 4x⁻¹ + (2x)².