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Let $p(x)$ be a monic polynomial of degree 6 such that $p(1) = 1,$ $p(2) = 2,$ $p(3) = 3,$ $p(4) = 4,$ $p(5) = 5,$ and $p(6) = 6.$ Find $p(7).$

 Jul 3, 2021
 #2
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Like so:

 

 Jul 4, 2021
 #3
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awesome! I got the six equations and six unknowns but was intimidated to solve it.

 Jul 4, 2021
 #4
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"I got the six equations and six unknowns but was intimidated to solve it."

 

I'm not surprised.  I wouldn't have bothered if I didn't have software available to do it!  Although there is a matrix capability on the web2.0 calculator on the home page here, it just goes up to 4x4 matrices, so wouldn't have helped here.  Probably Wolfram Alpha would do it, but I haven't tried it with this problem.

Alan  Jul 4, 2021
 #5
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You could also look up "method of differences" or "Lagrange interpolation" for faster computation. I'm not really familiar with it, but they both look faster than solving 6 simultaneous equations.

textot  Jul 4, 2021
 #6
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Thanks Alan,

Here is a matrix solver calculator.  Of course, you would  need to learn how to use it.,

 

https://www.symbolab.com/solver/matrix-calculator

 Jul 5, 2021

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