SOLUTION: How do I determine the polynomial function of least degree whose graph passes through the given points (−1, 2), (0, 0), (1, 1), (4, 58), p(x)=? Thank you!!!

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Question 896428: How do I determine the polynomial function of least degree whose graph passes through the given points (−1, 2), (0, 0), (1, 1), (4, 58), p(x)=?
Thank you!!!
Answer by josgarithmetic(39616) About Me  (Show Source):
You can put this solution on YOUR website!
One root of p is zero, so you can begin with p%28x%29=x%28ax%5E2%2Bbx%2Bc%29, to try to find a CUBIC polynomial.

The other three points let you form a system of three equations.

p%28x%29=ax%5E3%2Bbx%5E2%2Bcx=p%28x%29
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-a%2Bb-c=2, using (-1,2).

a%2Bb%2Bc=1, using (1,1).

a%2A4%5E3%2Bb%2A4%5E2%2Bc%2A4=58

64a%2B16b%2B4c=58, using point (4,58).

Solve for the coefficients a, b, c in the system
highlight_green%28system%28-a%2Bb-c=2%2Ca%2Bb%2Bc=1%2C64a%2B16b%2B4c=58%29%29



R2+R1, and R3+32*R1,



R3-20*R2
and one small division of R3 by constant, scaler



Which indicates highlight%28c=-11%29.
Using that continuing without further matrix operations,
highlight%28b=3%2F2%29 and highlight%28a=29%2F2%29.

Putting this all into expected cubic p(x),
highlight%28p%28x%29=%2829%2F2%29x%5E3%2B%283%2F2%29x%5E2-11x%29.
This was not checked but it should be checked.