# If polynomial P(x) with leading coefficient 1, of degree 4, is such that P(1)=1, P(2)=2, P(3)=3, P(4)=4. Then P(5)=? Note P(5) is not equal to 5

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P(x) = a x^4 + b x^3 + c x^2 + d x + e

P(1) = a + b + c + d + e = 1

P(2) = 16 a + 8 b + 4 c + 2 d + e = 2

P(3) = 81 a + 27 b + 9 c + 3 d + e = 3

P(4) = 256 a + 64 b + 16 c + 4 d + e = 4

P(5) = 625 a + 12 5 b + 25 c + 5 d + e = ??

Working on the four given equations, we get the following

15 a + 7 b + 3 c + d = 1

175 a + 37 b + 7 c+ d = 1

85 a + 21 b + 5 c + d = 1

40 a + 13 b + 4 c + d = 1

from the above four equations we get:

80 a + 15 b + 2 c = 0

45 a + 8 b + c = 0

Solving them we get 45 a - 80 a + c = 0

c = 35 a, b = -10 a, d = - 50 a, e = 1 + 24 a

P(5) = 625 a - 1250 a + 875 a - 250 a + 1 + 24 a

= 1 + 24 a = e = P(0)

P(5) = P(0)

P(1) = a + b + c + d + e = 1

P(2) = 16 a + 8 b + 4 c + 2 d + e = 2

P(3) = 81 a + 27 b + 9 c + 3 d + e = 3

P(4) = 256 a + 64 b + 16 c + 4 d + e = 4

P(5) = 625 a + 12 5 b + 25 c + 5 d + e = ??

Working on the four given equations, we get the following

15 a + 7 b + 3 c + d = 1

175 a + 37 b + 7 c+ d = 1

85 a + 21 b + 5 c + d = 1

40 a + 13 b + 4 c + d = 1

from the above four equations we get:

80 a + 15 b + 2 c = 0

45 a + 8 b + c = 0

Solving them we get 45 a - 80 a + c = 0

c = 35 a, b = -10 a, d = - 50 a, e = 1 + 24 a

P(5) = 625 a - 1250 a + 875 a - 250 a + 1 + 24 a

= 1 + 24 a = e = P(0)

P(5) = P(0)