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If p(x) = kx^2 - kx - 12, for all values of x, what is the value of k?

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If p(x) = kx^2 - kx - 12, for all values of x, what is the value of k?  [#permalink]

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New post Updated on: 06 Feb 2019, 06:51
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GMATH practice exercise (Quant Class 14)

If \(\,p\left( x \right) = k{x^2} - kx - 12\,\) for all values of \(x\), what is the value of the constant \(k\) ?


(1) \(p\left( 3 \right) = 0\)

(2) \(p(x)\) is divisible by \(x+2\)

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
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Originally posted by fskilnik on 06 Feb 2019, 06:48.
Last edited by Bunuel on 06 Feb 2019, 06:51, edited 1 time in total.
Renamed the topic.
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Re: If p(x) = kx^2 - kx - 12, for all values of x, what is the value of k?  [#permalink]

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New post 06 Feb 2019, 07:46
fskilnik wrote:
GMATH practice exercise (Quant Class 14)

If \(\,p\left( x \right) = k{x^2} - kx - 12\,\) for all values of \(x\), what is the value of the constant \(k\) ?


(1) \(p\left( 3 \right) = 0\)

(2) \(p(x)\) is divisible by \(x+2\)


At the very first look, you should be tempted towards D.

(1) \(p\left( 3 \right) = 0\)
Putting x as 3, the quadratic equation would convert into a linear equation with just one variable and so should be sufficient
Put x=3 so \(p\left( 3 \right) = 0.......\,p\left( 3 \right) =0= k{3^2} - k3 - 12\.......9k-3k=12...k=2\)
Sufficient

(2) \(p(x)\) is divisible by \(x+2\)
This should tell you to look if the equation can be converted in terms of x+2, and divisible by x+2 means substituting x as -2, we should get the equation as 0
\(k{x^2} - kx - 12\,=k(-2)^2-(-2)k-12=0........4k+2k-12=0...k=2\)
Sufficient

D
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Re: If p(x) = kx^2 - kx - 12, for all values of x, what is the value of k?  [#permalink]

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New post 06 Feb 2019, 10:44
fskilnik wrote:
GMATH practice exercise (Quant Class 14)

If \(\,p\left( x \right) = k{x^2} - kx - 12\,\) for all values of \(x\), what is the value of the constant \(k\) ?


(1) \(p\left( 3 \right) = 0\)

(2) \(p(x)\) is divisible by \(x+2\)


From 1
when we substitute the value as 3 in the expression \(\,p\left( x \right) = k{x^2} - kx - 12\,\)
We get a definite value for k

From 2
x + 2 means that x =-2, will satisfy the expression, which can be equated to 0
Again we will get a definite value for k

D
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Re: If p(x) = kx^2 - kx - 12, for all values of x, what is the value of k?  [#permalink]

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New post 07 Feb 2019, 07:05
fskilnik wrote:
GMATH practice exercise (Quant Class 14)

If the polynomial \(p\) in the variable \(x\) is defined by \(\,p\left( x \right) = k{x^2} - kx - 12\), for all values of \(x\), what is the value of the constant \(k\) ?

(1) \(p\left( 3 \right) = 0\)

(2) The polynomial \(p\) is divisible by \(x+2\)

Obs.: although the above version of the question stem is a bit "wordy", we believe it´s better (mathematically speaking).

\(p\left( x \right) = k{x^2} - kx - 12\,\,\,\,\,\left( * \right)\)

\(? = k\)

\(\left( 1 \right)\,\,\,0\,\, = \,\,p\left( 3 \right)\,\,\mathop = \limits^{\left( * \right)} \,\,k{\left( 3 \right)^2} - k\left( 3 \right) - 12\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,k\,\,{\rm{unique}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\rm{SUFF}}.\)

\(\left( 2 \right)\,\,\,0 = p\left( { - 2} \right)\,\,\mathop = \limits^{\left( * \right)} \,\,k{\left( { - 2} \right)^2} - k\left( { - 2} \right) - 12\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,k\,\,{\rm{unique}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\rm{SUFF}}{\rm{.}}\)


The correct answer is therefore (D).

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Our high-level "quant" preparation starts here: https://gmath.net
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Re: If p(x) = kx^2 - kx - 12, for all values of x, what is the value of k?   [#permalink] 07 Feb 2019, 07:05
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