Question

when x3 -2x2 +ax -b is divided by x2 +2x -3 the reminder is x -6the value of a and b are respectively?​

Answer 1

please see the attachment

Answer 2

Answer:

a = -2 and b = 6

Step-by-step explanation:

It's given that x³ - 2x² + ax - b when divided by x² - 2x - 3 leaves remainder as x - 6 .

Now if we subtract the dividend from the remainder we have a polynomial exactly divisibe by x² - 2x - 3 .

So , x³ - 2x² + ax - b

- x + 6

-----------------------

x³ - 2x² + ax - x - b+6

=> x³ - 2x² + x(a - 1) + (6 - b)

Now lets factorize x² - 2x - 3

=> x² - 3x + x - 3

=> x(x - 3) + 1(x - 3)

=> (x - 3)(x + 1)

Thus x - 3 and x + 1 are factors of

x³ - 2x² + x(a - 1) + (6 - b) also 3 and -1 are zeroes of x³ - 2x² + x(a - 1) + (6 - b) .

Lets substitute the values to find a and b.

f(-1) = (-1)³ - 2(-1)² - (a - 1) + (6 - b) = 0

=> -1 -2 - (a - 1) + (6 - b) = 0

=> - (a - 1) + (6 - b) = 3

=> -a + 1 + 6 - b = 3

=> -a - b = -4

=> a + b = 4...(i)

Also equate for f(3)

f(3) = (3)³ - 2(3)² +3(a - 1) + (6 - b) = 0

=> 27 - 18 + 3a - 3 + 6 - b = 0

=> 3a - b + 12 = 0

=> 3a - b = -12...(ii)

Now from eq (i) we have

a = 4 - b

So we put this value in eq (ii)

3(4 - b) - b = -12

12 - 3b - b = -12

-4b = -24

Thus b = 6

Now substitute this value in eq (i)

a + 6 = 4

Thus a = -2

Therefore a = -2 and b = 6

Thanks mark as brainliest if it helped you.