Not So Big!
+ + + ( a − b ) ( a − c ) ( a − d ) ( x − b ) ( x − c ) ( x − d ) ( b − c ) ( b − d ) ( b − a ) ( x − c ) ( x − d ) ( x − a ) ( c − d ) ( c − a ) ( c − b ) ( x − d ) ( x − a ) ( x − b ) ( d − a ) ( d − b ) ( d − c ) ( x − a ) ( x − b ) ( x − c ) = ?
Hint : An equation of n th degree with more than n roots is an identity. This is studied extensively in the wiki: The method of undetermined coefficients .
The answer is 1.
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3 solutions
Ha ha ,nice use of that hint.
Is it a way to solve these questions or it just worked this time
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Because the question does not state any value of x (or any of the variables), and we know the answer is an integer, we can substitute any number in.
Since the answer is not a variable, simply put x=a, b, c or d. And get the answer in seconds!!!!
Let f ( x ) denote the given expression such that degree of f ( x ) is 3 .
Then we can see f ( x ) = 1 for x = a , b , c , d or alternatively f ( x ) − 1 = 0 has four roots.
An n t h degree equation can have more than n roots only if it's an identity !!
Therefore f ( x ) − 1 = 0 ∀ x ∈ R or f ( x ) = 1 ∀ x ∈ R .