Inspired by Mehul Arora
The function f ( x ) is a monic quartic polynomial satisfying f ( 1 ) = 1 , f ( 2 ) = 4 , f ( 3 ) = 9 , f ( 4 ) = 1 6 . What is f ( 5 ) ?
The answer is 49.
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2 solutions
@Swapnil Das Can you add a link to the inspiration? :P
Because I personally do not remember this sort of a question that I posted :P
It's f ( x ) .
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Haha, yes. Good observation.
It doesn't matter whether you represent a function with an I or a P or a G.
But why can't the ( x − 1 ) ( x − 2 ) ( x − 3 ) ( x − 4 ) part be divided with any integer.That would still satisfy the conditions and f ( 5 ) would also change that way.
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It says it's Monic which means it's leading coefficient is 1.
How you found that function ? (and someone can give me link what is quartic monic polynomial ? I didn't found it anywhere)
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You must study polynomials and basic definitions. Thanks!
Values: 1, 4, 9, 16, ?
First differences: 3, 5, 7, ?
Second differences: 2, 2, ?
Third differences, 0, ?
For a quartic polynomial, the fourth differences must be equal to 4 ! ⋅ a , where a is the coefficient of x 4 . For a monic polynomial, a = 1 so that the fourth differences must be 24.
Working from bottom to top,
Third differences, 0, 24
Second differences: 2, 2, 26
First differences: 3, 5, 7, 33
Values: 1, 4, 9, 16, 49
Sir, In the second difference we have got our constant number - 2. Doesn't our process ends here ? Then by backward approach i have got f ( 5 ) = 2 5 . But this isn't a corresponding answer(49) . Please correct me . I think i have missed out something in the learning of the Method of diffrences. Please point me out where i have missed out.
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In the second differences there are two values that are equal , but that does not mean it should stay constant .
The problem states that the polynomial is monic quartic , so its degree is four and the leading coefficient is one ( a 4 = 1 ): p ( x ) = x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 .
In general, for an n -th degree polynomial, the n -th difference is equal to n ! a n . In this case, it should be 1 ⋅ 4 ! = 2 4 .
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Sir , please give me a link where all this important data is written such as - In general, for an - n th degree polynomial, the n -th difference is equal to n ! a n .
P ( x ) = ( x − 1 ) ( x − 2 ) ( x − 3 ) ( x − 4 ) + x 2
P ( 5 ) = 4 ! + 5 2 = 4 9