Verified answer

Answer:

a = 5, b = 8, p(2) = -2

Step-by-step explanation:

According to Remainder Theorem:

If we divide a polynomial p(x) by (x-a) then the remainder is equal to p(a).

Given polynomial:

p(x)=x²-2x³+3x²-ax+b

Dividing it by (x-1) will give remainder equivalent to p(1) which is equal to 5.

Thus,

p(1) = 5

(1)²-2(1)³+3(1)²-a(1)+b = 5

1 - 2 + 3 - a + b = 5

1 + 3 - 2 + b - a = 5

2 + b - a = 5

b - a = 5 - 2

b - a = 3     . . . . equation(1)

Dividing it by (x+1) will give remainder equivalent to p(-1) which is equal to 19.

Thus,

p(-1) = 19

(-1)²-2(-1)³+3(-1)²-a(-1)+b = 19

1 -2(-1)+3(1)+a+b = 19

1 + 2 + 3 + a + b = 19

6 + a + b = 19

a + b = 19 - 6

a + b = 13     . . . . equation(2)

Adding equation(1) and equation(2),

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

(b + b) + (a - a) = 16

2b = 16

b = 16/2

b = 8

Substituting value of b in equation(1),

8 - a = 3

a = 8 - 3

a = 5

Thus given polynomial becomes,

p(x)=x²-2x³+3x²-5x+8

Dividing p(x) by (x-2) will give remainder equivalent to p(2).

p(2) = (2)² - 2(2)³ + 3(2)² - 5(2) + 8

p(2) = 4 - 2(8) + 3(4) - 10 + 8

p(2) = 4 - 16 + 12 - 10 + 8

p(2) = 4 + 12 + 8 - 16 -10

p(2) = 24 - 16 - 10

p(2) = 24 - 26

p(2) = -2