Q. 74.1( 12 Votes )

If the polynomial x4 – 2x3 + 3x2 – ax + b is divided by (x – 1) and (x + 1) and the remainders are 5 and 19 respectively. But if that polynomial is divided by x + 2, then what will be the remainder —Let us calculate.

Answer :

We have the polynomial x4 – 2x3 + 3x2 – ax + b.


Let it be P(x), such that


P(x) = x4 – 2x3 + 3x2 – ax + b …(i)


According to the question,


P(x) is divided by (x – 1) and (x + 1), and leaves the remainder 5 and 19 respectively.


We will use remainder theorem here.


By Remainder theorem that says, f(x) is a polynomial of degree n (n ≥ 1) and ‘a’ is any real number. If f(x) is divided by (x – a), then the remainder will be f(a).


At first, let’s find the zero of the linear polynomial, (x – 1).


To find zero,


x – 1 = 0


x = 1


Now, let’s find the zero of the linear polynomial, (x + 1).


To find zero,


x + 1 = 0


x = -1


From Remainder theorem, we can say


When P(x) is divided by (x – 1), the remainder comes out to be 5.


We can also say that,


The remainder comes out to be P(1).


P(1) = 5


(1)4 – 2(1)3 + 3(1)2 – a(1) + b = 5


1 – 2 + 3 – a + b = 5


b – a + 2 = 5


b – a = 5 – 2


b – a = 3 …(ii)


And when P(x) is divided by (x + 1), the remainder comes out to be 19.


We can also say that,


The remainder comes out to be P(-1)


P(-1) = 19


(-1)4 – 2(-1)3 + 3(-1)2 – a(-1) + b = 5


1 + 2 + 3 + a + b = 5


a + b + 6 = 5


a + b = 5 – 6


a + b = -1 …(iii)


Solving equations (ii) and (iii), we get


(b – a) + (a + b) = 3 + (-1)


2b = 3 – 1


2b = 2



b = 1


Putting b = 1 in equation (ii),


b – a = 3


1 – a = 3


a = 1 – 3


a = -2


The values of a = -2 and b = 1.


The polynomial = x4 – 2x3 + 3x2 – ax + b


The polynomial = x4 – 2x3 + 3x2 – (-2)x + 1


The polynomial = x4 – 2x3 + 3x2 + 2x + 1


So, when polynomial x4 – 2x3 + 3x2 + 2x + 1 is divided by (x + 2), the remainder can be calculated by using Remainder theorem.


First, we need to find zero of the linear polynomial, (x + 2).


To find zero,


Put (x + 2) = 0


x = -2


So, Required remainder = P(-2)


Required remainder = (-2)4 – 2(-2)3 + 3(-2)2 + 2(-2) + 1


Required remainder = 16 + 16 + 12 – 4 + 1


Required remainder = 32 + 12 – 3


Required remainder = 44 – 3


Required remainder = 41


Thus, remainder is 41.


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