Question

If f(x) = 2x^ 6 – 5x – 1, What is the remainder when f(x) is divided by x^2?

Option 1: 2x^4 - 5x - 1
Option 2: 2x^4 - 5
Option 3: -5x - 1
Option 4: -5

Answer 1

The remainder when the polynomial f(x) = 2x^6 - 5x - 1 is divided by x^2 is the polynomial of degree less than 2, which is -5x - 1 (Option 3).

Step-by-step explanation:

The question asks for the remainder when the polynomial f(x) = 2x^6 − 5x − 1 is divided by x^2. According to the Division Algorithm for polynomials, when a polynomial f(x) is divided by a divisor d(x) where the degree of d(x) is lower than that of f(x), the remainder r(x) will have a degree less than the degree of d(x). In this case, d(x) = x^2, so the remainder r(x) must be a polynomial of degree less than 2.

The possible remainders given as options have various degrees, but we're looking for the one with degree less than 2. This immediately eliminates Options 1 and 2, as they contain terms with x raised to the power of 4. Looking at Options 3 and 4, we can see that Option 3, −5x − 1, is the correct answer since it is a polynomial of degree 1, which is less than 2 (the degree of x^2).