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21 Jan 2022, 05:45
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
65% (02:07) correct
35%
(02:25)
wrong
based on 177
sessions
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For all positive integers x, \(x^3+ax^2+8x+5\) is a multiple of \((x+1)\). What is the value of a?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 7
(A) 2
(B) 3
(C) 4
(D) 5
(E) 7
05 Dec 2023, 01:49
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Bunuel
This is a straightforward application of Factor theorem. In simple words, (x+1) is a factor of the cubic, so when we multiply (x+1) with a quadratic, we will get the cubic equation. It will look something like this:
Given:
\(x^3+ax^2+8x+5 = (x+1)(x^2 ...) \)
If x were -1, then (x+1) = 0 and hence, \(x^3+ax^2+8x+5 = 0 * (x^2 ...) = 0\)
Put x = -1, \((-1)^3+a(-1)^2+8(-1)+5 = 0\)
a = 4
Answer (C)
General Discussion
24 Jan 2022, 01:15
Kudos
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Dividend = \(x^3+ax^2+8x+5\) and Divisor =\((x+1)\). We need to find out what multiplies \((x+1)\) to obtain the \(x^3+ax^2+8x+5\).
The co-efficient of\( x^0\) in the dividend is \(5\). We can obtain \(5\) from \((x+1)\) only by multiplying \((x+1) \)by \(5\).
We get, \(5(x+1) = 5x+5\).
Dividend has \(8x\) and our product of \(5(x+1)\) has given us \(5x\). The difference of \(3x\) will be obtained by multiplying \((x+1)\) by \(3x\).
\(3x*(x+1) = 3x^2 + 3x.\)
The \(x^2\) term in dividend has coefficient of \('a'\) and our product of \(3x*(x+1)\) has provided coefficient of \(3\).
The remaining \((a-3)\) will be obtained by multiplying \((x+1)\) by \(x^2(a-3)\).
\((x+1)* x^2 * (a-3) = (x^3+x^2)(a-3) = (a-3)x^3 + (a-3)x^2\).
Thus we get, \((x+1) (x^2(a-3) + 3x + 5) = x^3+ax^2+8x+5\)
The co-efficient of \(x^3\) in resultant product is \((a-3)\) and the co-efficient of \(x^3\) is given as \(1\).
=> \(a-3 = 1\)
=> \(a = 4\).
Answer: C
The co-efficient of\( x^0\) in the dividend is \(5\). We can obtain \(5\) from \((x+1)\) only by multiplying \((x+1) \)by \(5\).
We get, \(5(x+1) = 5x+5\).
Dividend has \(8x\) and our product of \(5(x+1)\) has given us \(5x\). The difference of \(3x\) will be obtained by multiplying \((x+1)\) by \(3x\).
\(3x*(x+1) = 3x^2 + 3x.\)
The \(x^2\) term in dividend has coefficient of \('a'\) and our product of \(3x*(x+1)\) has provided coefficient of \(3\).
The remaining \((a-3)\) will be obtained by multiplying \((x+1)\) by \(x^2(a-3)\).
\((x+1)* x^2 * (a-3) = (x^3+x^2)(a-3) = (a-3)x^3 + (a-3)x^2\).
Thus we get, \((x+1) (x^2(a-3) + 3x + 5) = x^3+ax^2+8x+5\)
The co-efficient of \(x^3\) in resultant product is \((a-3)\) and the co-efficient of \(x^3\) is given as \(1\).
=> \(a-3 = 1\)
=> \(a = 4\).
Answer: C
