Question

if a polynomial is defined as P(x) = 2x​5+ ax4+ bx3+ cx2+ dx + e such that​P(0) = 4,​P(1) = 5,​P(2) = 8,
P(3) = 13,P(4) = 20 find the value ofP(5).

Answer 1

P(0) = 4  => e = 4

P(1) = 5  => a+b+c+d = -1       
P(2) = 8  =>  8a + 4b + 2 c + d = -30
         =>  7a + 3b + c = -29
P(3) = 13 =>  27 a + 9 b + 3c +d = -159
         =>  13 a + 4 b + c = -79
     =>  6a + b = -50

P(4) = 20  =>   64 a + 16 b + 4c + d = -508
               =>  21a + 5 b + c = -169
           => 7a + b = -70
     => a = -20  and    b = 70

then c = -99 and  d = 48

P(5) = 6250 - 12500 + 8750 - 15 * 99 + 4  = 29