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02 Mar 2020, 05:42
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
35% (02:17) correct
65%
(02:25)
wrong
based on 157
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If \((x + 1)(x – 2)(x + 3)(x – 4)(x + 5)…(x – 100) = a_0 + a_1x + a_2x^2+… +a_{99}x^{99}+ a_{100}x^{100}\) then the value of \(a_{99}\) is equal to:
A. 100
B. 50
C. 0
D. -50
E. -100
Are You Up For the Challenge: 700 Level Questions
A. 100
B. 50
C. 0
D. -50
E. -100
Are You Up For the Challenge: 700 Level Questions
02 Mar 2020, 06:08
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Bunuel
\((x + 1)(x – 2)(x + 3)(x – 4)(x + 5)…(x – 100) = a_0 + a_1x + a_2x^2+… +a_{99}x^{99}+ a_{100}x^{100}\)
\(a_{99}\) is the coefficient of second highest power of x, Let's use Induction
\((x + 1)(x – 2) = x^2 - x - 2\) has coefficient of second highest power of x (i.e. x) as -1 i.e. 1-2
\((x + 1)(x – 2)(x+3) = x^3 +2x^2 - 5x - 6\) has coefficient of second highest power of x (i.e. \(x^2\)) as 2 i.e. 1-2+3
Similarlly
\(a_{99} = 1-2+3-4+5-6+..... = -1 + -1 + -1 +.... 50 times = -50\)
Answer: Option D
General Discussion
06 Apr 2022, 01:04
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General Property:
For any Polynomial of the following form:
a (x)^n + b (x)^(n-1) + c (x)^(n-2) …… y (x)^1 + K
Where a, b, c, and y are Coefficients of each of the Variable Terms
And
Where K = constant value
The SUM OF ROOTS can always be found by:
(-)b
___
a
Which is the negation of the coefficient in front of the 2nd Highest Degree term divided by the coefficient in front of the Highest degree term.
In the question, we are given the fully factored form on the left side and multiplied out Polynomial on the right side
the Coefficient denoted by A(99) will represent:
(-) (Sum of Roots)
_______________
A(100)
The roots are the values that make the polynomial equal zero, these are
-1 + 2 - 3 + 4 - 5 + 6 ……. -97 + 98 - 99 + 100
Which works out to be =
+50
Then to find A(99) we take this negated value and divide it by the coefficient that comes in front of the Term with the highest degree: since in each factor, the coefficient is +1, A(100) will work out to be = +1
Answer:
(-) (50)
______
1
-50
*D*
Posted from my mobile device
For any Polynomial of the following form:
a (x)^n + b (x)^(n-1) + c (x)^(n-2) …… y (x)^1 + K
Where a, b, c, and y are Coefficients of each of the Variable Terms
And
Where K = constant value
The SUM OF ROOTS can always be found by:
(-)b
___
a
Which is the negation of the coefficient in front of the 2nd Highest Degree term divided by the coefficient in front of the Highest degree term.
In the question, we are given the fully factored form on the left side and multiplied out Polynomial on the right side
the Coefficient denoted by A(99) will represent:
(-) (Sum of Roots)
_______________
A(100)
The roots are the values that make the polynomial equal zero, these are
-1 + 2 - 3 + 4 - 5 + 6 ……. -97 + 98 - 99 + 100
Which works out to be =
+50
Then to find A(99) we take this negated value and divide it by the coefficient that comes in front of the Term with the highest degree: since in each factor, the coefficient is +1, A(100) will work out to be = +1
Answer:
(-) (50)
______
1
-50
*D*
Posted from my mobile device
