After several operations of differentiation and multiplication by (x+1) performed in an arbitrary order, the poylnomial (x8+x7) is changed to (ax+b). Show that |(a-b)| is divisible by 49.....
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1 Answers
1The repeated differentiation of (x8+x7) will give 8!x + 7!
Multiply this by (x+1) and you get 8! x2 + 9X7! x + 7!
Differentiate this again to get 2X8! x + 9X7!
Here a = 2 x 8! = 16 x 7! and b = 9 x 7!
a - b = 7 x 7! = 49 x 6!
So it is divisible by 49.
No matter how many times you multiply by x + 1 and differentiate the answer, there will always be an 8! in the coefficient of the x term (a) and a 7! in the coefficient of the constant term (b) after repeated differentiation and multiplication. Try it out. The solution that i gave is only for multiplication once, and repeated differentiation.
Hence, when you subtract a-b, then you get k x 8! - k x 7! where k is constant (because when we differentiate xn we get nxn-1 so each time differentiation is carried out, the powers will keep getting multiplied. Supposing that we have multiplied (x+1) many times then the number '2' will keep getting multiplied)
So we get k x 8 x 7! - k x 7! = (8k-k) x 7!
Hence the resulting answer should be divisible by 49.