## If polynomial x4 − 2ax3 + 5×2– 3x + 1 is divided by x + 2, it leaves 3 as remainder, then find the value of ‘a’​

Question

If polynomial x4 − 2ax3

+ 5×2– 3x + 1 is divided by x + 2, it leaves 3 as remainder, then find the value of ‘a’​

in progress 0
1 month 2021-09-22T21:27:29+00:00 1 Answer 0 views 0

1.

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.

Done!

HOPE THIS HELPS…☺☺☺