Let-P-x-is-a-polynomial-such-that-P-1-1-P-2-2-P-3-3-and-P-4-5-Find-the-value-of-P-6-

Question Number 19638 by Tinkutara last updated on 13/Aug/17

Let P (x) is a polynomial such that

P (1) = 1, P (2) = 2, P (3) = 3, and

P (4) = 5 . Find the value of P (6) .

Answered by ajfour last updated on 13/Aug/17

let P (x) = {ax}^{3} + {bx}^{2} + cx + d

P (1) = 1, \Rightarrow a + b + c + d = 1

P (2) = 2, \Rightarrow 8 a + 4 b + 2 c + d = 2

P (3) = 3, \Rightarrow 27 a + 9 b + 3 c + d = 3

P (4) = 5, \Rightarrow 64 a + 16 b + 4 c + d = 5

\Rightarrow 37 a + 7 b + c = 2

19 a + 5 b + c = 1

7 a + 3 b + c = 1

\Rightarrow 18 a + 2 b = 1

12 a + 2 b = 0

\Rightarrow 6 a = 1 or a = \frac{1}{6}

b = - 1, c = 1 - 3 b - 7 a

\Rightarrow c = 1 + 3 - \frac{7}{6} = \frac{17}{6}

d = 1 - a - b - c

= 1 - \frac{1}{6} + 1 - \frac{17}{6} = - 1

P (x) = \frac{x^{3}}{6} - x^{2} + \frac{17 x}{6} - 1

P (6) = 36 - 36 + 17 - 1 = 16 .

Commented by Tinkutara last updated on 13/Aug/17

Thank you very much Sir!