Q. 114.1( 37 Votes )

# The Polynomial f(x) = x^{4} -2x^{3 }+3x^{2 }-ax + b when divided by (x-1) and (x+1) leaves the remainders 5 and 19 respectively. Find the values of a and b. Hence, find the remainder when f(x) is divided by (x-2).

Answer :

Let f(x) = x^{4} – 2x^{3} + 3x^{2} – ax + b

Now,

f(1) = 1^{4} – 2(1)^{3} + 3(1)^{2} – a(1) + b

5 = 1 – 2 + 3 – a + b

3 = - a + b (i)

And,

f(-1) = (-1)^{4} – 2(-1)^{3} + 3(-1)^{2} – a(-1) + b

19 = 1 + 2 + 3 + a + b

13 = a + b (ii)

Now,

Adding (i) and (ii),

8 + 2b = 24

2b = 16

b = 8

Now,

Using the value of b in (i)

3 = - a + 8

a = 5

Hence,

a = 5 and b = 8

Hence,

f(x) = x^{4} – 2(x)^{3} + 3(x)^{2} – a(x) + b

= x^{4} – 2x^{3} + 3x^{2} – 5x + 8

f(2) = (2)^{4} – 2(2)^{3} + 3(2)^{2} – 5(2) + 8

= 16 – 16 + 12 – 10 + 8

= 20 – 10

= 10

Therefore, remainder is 10

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