Answer:

a = 5

b = 8

Remainder = 10

Step-by-step explanation:

There is theorem known as “Polynomial Remainder Theorem” or “ Bezout’s Theorem”. It is Stated as -

A Polynomial f(x) if divided by a linear polynomial (x-a) LEAVES remainder which equals f(a).

So , getting BACK to our question -

f(x) = x^4 - 2x^3 + 3x^2 - ax + b

So , when it is divided by (x - 1) it’ll leave a remainder = f(1) = 5 (Given).

f(1) = 1^4 - 2×1^3 + 3×1^2 - a×1 + b = 5

=> 1 - 2 + 3 - a + b = 5

=> a - b = (-3) …. Eqn(1)

Now , Similarly -

f(-1) = (-1)^4 - 2×(-1)^3 + 3×(-1)^2 - a×(-1) + b = 19

=> 1 + 2 + 3 + a + b = 19

=> a + b = 13 …. Eqn(2)

Now , adding equations (1) and (2) , We’ll get -

(a+b) + (a-b) = (-3) + 13

=> 2a = 10 => a = 5

So , (a +b) = 13 IMPLIES b = 8

Hence , Values of a and b are 5 and 8 respectively.

Hence, our

p(x) = x^4 - 2x^3 + 3x^2 - 5x + 8

Now,

Dividend p(x) = x^4 - 2x^3 + 3x^2 - 5x + 8

Divisor G(x) = x - 2

By using remainder theorem, we get,

x - 2 = 0

=> x = 2

By APPLYING the values of x in p(x), we get,

(2)^4 - 2×(2)^3 + 3×(2)^2 - (5×2) + 8

=> 16 - 16 + 12 - 10 + 8

=> 2 + 8

=> 10

Hence, the remainder r(x), when x^4 - 2x^3 + 3x^2 - 5x + 8 is divided by (x-2) is 10