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A monic polynomial P of degree 5 satisfies P(1)=1, P(2)=4, P(3)=9, P(4)=16, P(5)=73. Find P(6).

 Dec 27, 2021
 #1
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P(6) = 122.

 Dec 27, 2021
 #2
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P(x) = x^2 for when x = {1, 2, 3, 4}. 

F(x) = P(x) - x^2

F(5) = 73 - 25

F(5) = 48

Since P(x) is a monic polynomial with a degree of 5, so is F(x).

F(x) = 0 for when x = {1, 2, 3, 4, 5}

F(x) = (x-a)(x-1)(x-2)(x-3)(x-4)

F(5) = 24(5 - a)

a = 3

F(x) = (x-3)(x-1)(x-2)(x-3)(x-4)

(x-3)(x-1)(x-2)(x-3)(x-4) = P(x) - x^2

P(x) = (x-3)(x-1)(x-2)(x-3)(x-4) + x^2

P(6) = 396

 

Did I get it right?

 

=^._.^=

 Dec 28, 2021
 #3
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yep! good job

Guest Dec 31, 2021

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