Use long division to find the quotient and remainder when 2x5 + 4x4 − x3 − x2 + 5 is divided by 2x2 − 1.

Sure, here is the step-by-step solution:

Step 1: Arrange the dividend and the divisor in descending order of their degrees.

Dividend: 2x^5 + 4x^4 - x^3 - x^2 + 5 Divisor: 2x^2 - 1

Step 2: Divide the first term of the dividend by the first term of the divisor. This will give you the first term of the quotient.

First term of the quotient: 2x^5 / 2x^2 = x^3

Step 3: Multiply the divisor by the first term of the quotient and subtract the result from the dividend.

(2x^2 - 1) * x^3 = 2x^5 - x^3

Subtract this from the dividend:

2x^5 + 4x^4 - x^3 - x^2 + 5 - (2x^5 - x^3) = 4x^4 - x^2 + 5

Step 4: Repeat steps 2 and 3 with the new dividend.

First term of the quotient: 4x^4 / 2x^2 = 2x^2

(2x^2 - 1) * 2x^2 = 4x^4 - 2x^2

Subtract this from the new dividend:

4x^4 - x^2 + 5 - (4x^4 - 2x^2) = x^2 + 5

Step 5: Repeat steps 2 and 3 with the new dividend.

First term of the quotient: x^2 / 2x^2 = 1/2

(2x^2 - 1) * 1/2 = x^2 - 1/2

Subtract this from the new dividend:

x^2 + 5 - (x^2 - 1/2) = 5 + 1/2 = 5.5

Step 6: Since the degree of the new dividend (0) is less than the degree of the divisor (2), we stop here. The last dividend is the remainder.

So, the quotient is x^3 + 2x^2 + 1/2 and the remainder is 5.5.