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# Polynomials!

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1. Suppose g(x) is a polynomial of degree five for which g(1) = 2, g(2) = 3, g(3) = 4, g(4) = 5, g(5) = 6, and g(6) = -113. Find g(0).

2. Suppose f(x) is a polynomial of degree 4 or greater such that f(1)=2, f(2)=3, and f(3)=5. Find the remainder when f(x) is divided by (x-1)(x-2)(x-3).

Dec 15, 2018

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I know how to do the first one.....but.....not the second

1. Suppose g(x) is a polynomial of degree five for which g(1) = 2, g(2) = 3, g(3) = 4, g(4) = 5, g(5) = 6, and g(6) = -113. Find g(0).

The first one sets up a  system of 6 equations with 6 unknowns.....this is very tedious to solve by hand.....so....I'm not!!!....I'll let this website do the "heavy lifting" :

https://matrix.reshish.com/gauss-jordanElimination.php

Here is the system :

a + b + c + d + e + f = 2

32a + 16b + 8c + 4d + 2e + f = 3

243a + 81b + 27c + 9d + 3e + f = 4

1024a + 256b + 64c + 16d + 4e + f = 5

3125a + 625b + 125c + 25d + 5e + f  = 6

7776a + 1296b + 216c + 36d + 6e + f = -113

We are only interested in the value of "f"   = 121

And this is   =   f(0)

BTW....the polynomial is

-x^5  + 15x^4 - 85x^3 + 225x^2 - 273x + 121   Dec 15, 2018