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25 Nov 2021, 03:30
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A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
57% (02:16) correct
43%
(02:28)
wrong
based on 176
sessions
History
Date
Time
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If \(P(x)\) be a polynomial satisfying the identity \(P(x^2) +2x^2 +10x = 2xP(x+1) +3\), then \(P(x)\) is
(A) \(2x+3\)
(B) \(3x-4\)
(C) \(3x+2\)
(D) \(2x–3\)
(E) \(3x–3\)
(A) \(2x+3\)
(B) \(3x-4\)
(C) \(3x+2\)
(D) \(2x–3\)
(E) \(3x–3\)
02 Dec 2021, 05:13
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A look at the answer choices confirms that the answer is of the form ax + b.
So, let us assume that P(x) = ax + b.
Thus
P(x) = ax + b
\(P(x^2) = ax^2 + b\)
\(P(x+1) = ax + a + b\)
Substitute these 3 statements in the question.
\(ax^2 + 2x^2 + 10x + b\) = \(2ax^2 + 2ax +2bx + 3\)
This gives us b = 3, and a = 2.
Thus Option a is the answer
So, let us assume that P(x) = ax + b.
Thus
P(x) = ax + b
\(P(x^2) = ax^2 + b\)
\(P(x+1) = ax + a + b\)
Substitute these 3 statements in the question.
\(ax^2 + 2x^2 + 10x + b\) = \(2ax^2 + 2ax +2bx + 3\)
This gives us b = 3, and a = 2.
Thus Option a is the answer
General Discussion
02 Dec 2021, 01:41
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Looking at both the left and the right side one sees that there is a constant of positive 3 on the right side. For these the two sides to be equal there needs to be a balance. Looking at the left side, the only way that there will be a constant of 3 is if the function itself contains positive 3. The only answer choice which has this is A.
Testing answer choice A: Plugging in \(2x + 3\) into \(P(x^2) + 2x^2 + 10x = 2xP(x+1) + 3\)
\(2x^2 + 3 + 2x^2 + 10x = 2x*[2(x+1) + 3] + 3\)
\(4x^2 + 10x + 3 = 2x*[2x + 2 + 3] + 3\)
\(4x^2 + 10x + 3 = 4x^2 + 10x +3\)
Both sides are equal to one another.
Answer A
Testing answer choice A: Plugging in \(2x + 3\) into \(P(x^2) + 2x^2 + 10x = 2xP(x+1) + 3\)
\(2x^2 + 3 + 2x^2 + 10x = 2x*[2(x+1) + 3] + 3\)
\(4x^2 + 10x + 3 = 2x*[2x + 2 + 3] + 3\)
\(4x^2 + 10x + 3 = 4x^2 + 10x +3\)
Both sides are equal to one another.
Answer A
