Question

the remainder when x^5-x^4-5x^3+3x^2-x+8 is divided by 2x-1​

Answer 1

Answer:

the remainder when x^5-x^4-5x^3+3x^2-x+8 is divided by 2x-1

Answer 2

Step-by-step explanation:

Let p(x) be any polynomial of degree greater than or equal to one and let 'a' be any real number.

If a polynomial p(x) is divided by x - a then the remainder is p(a).

Let p(x) = x3 + 3x2 + 3x + 1

(i) The root of x + 1 = 0 is -1

p(-1) = (-1)3 + 3(-1)2 + 3(-1) + 1

= -1 + 3 - 3 + 1

= 0

Hence by the remainder theorem, 0 is the remainder when x3 + 3x2 + 3x + 1 is divided by x + 1. We can also say that x + 1 is a factor of x3 + 3x2 + 3x + 1.

(ii) The root of x - (1/2) = 0 is 1/2.

p(1/2) = (1/2)3 + 3(1/2)2 + 3(1/2) + 1

= 1/8 + 3/4 + 3/2 + 1

= (1 + 6 + 12 + 8)/8 = 27/8

Hence by the remainder theorem, 27 / 8 is the remainder when x3 + 3x2 + 3x + 1 is divided by x.