samuelclarke
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- 2012
I need help with this question. Can someone show the worked out solution? thanks!
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HELP POLYNOMIALS (1 Viewer)
- Thread starter samuelclarke
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I'm on iPhone, but all you is equate the two.
Then you take one of th polynomials to the other
Side and them you factorise the like powers and then
You state this can only equal zero only if the coefs of
Each equals zero and you get the reunified answe
Then you take one of th polynomials to the other
Side and them you factorise the like powers and then
You state this can only equal zero only if the coefs of
Each equals zero and you get the reunified answe
To expand on Math Man's answer, the proof of this theorem depends on an important corollary: A polynomial with degree of at most n with more then n roots must be the zero polynomial.
With this the proof is simple and was outlined above:
We form a new polynomial, G(z) say, such that
G(z) = p1(z) - p2(z)
now it is given that p1(z) and p2(z) are equal at more than n values. lets call these points q1,q2,...qk where k>n
Obviously G(q1)=G(q2)=...=G(qk)=0,
ie, G has more than n roots but since G = p1 - p2, it is of most n degree. So by our corollary we conclude G is the zero polynomial
or that p1(z) - p2(z) = 0 or p1(z) = p2(z).
With this the proof is simple and was outlined above:
We form a new polynomial, G(z) say, such that
G(z) = p1(z) - p2(z)
now it is given that p1(z) and p2(z) are equal at more than n values. lets call these points q1,q2,...qk where k>n
Obviously G(q1)=G(q2)=...=G(qk)=0,
ie, G has more than n roots but since G = p1 - p2, it is of most n degree. So by our corollary we conclude G is the zero polynomial
or that p1(z) - p2(z) = 0 or p1(z) = p2(z).