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10 Mar 2016, 05:10
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E
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:
58% (01:30) correct
42%
(01:57)
wrong
based on 271
sessions
History
Date
Time
Result
Not Attempted Yet
How many roots does x^6 –12x^4 + 32x^2 = 0 have?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
Updated on: 29 Mar 2017, 08:58
Originally posted by davedekoos on 10 Mar 2016, 14:28.
Last edited by davedekoos on 29 Mar 2017, 08:58, edited 1 time in total.
Last edited by davedekoos on 29 Mar 2017, 08:58, edited 1 time in total.
Kudos
Bookmarks
To find the roots we can factor the polynomial down to first order expressions and count how many roots we get.
The first thing to do is to factor out \(x^2\)
Then we have \(x^2(x^4-12x^2+32)=0\)
Now to make things more familiar we can substitute \(y=x^2\)
Now it looks like this
\(y(y^2-12y+32)=0\)
And we can factor it like we are used to
\(y(y-4)(y-8)=0\)
y=0
y=4
y=8
So substituting \(x^2\) back in for y:
\(x^2=0\)
\(x^2=4\)
\(x^2=8\)
So then the roots are
\(x=0\)
\(x=-2\)
\(x=2\)
\(x=-\sqrt{8}\)
\(x=\sqrt{8}\)
5 roots. Answer: E
Note, since we are not asked to find the actual roots, we only need to determine the number of roots. After the first step of factoring out the \(x^2\), we have \(x^2\) and a 4th order expression. A 4th order expression will have 4 roots (unless it's a perfect square, but our expression is not a perfect square). So the total number of roots will be 4 from the 4th order expression, and one from the \(x^2\). Total roots = 5.
The first thing to do is to factor out \(x^2\)
Then we have \(x^2(x^4-12x^2+32)=0\)
Now to make things more familiar we can substitute \(y=x^2\)
Now it looks like this
\(y(y^2-12y+32)=0\)
And we can factor it like we are used to
\(y(y-4)(y-8)=0\)
y=0
y=4
y=8
So substituting \(x^2\) back in for y:
\(x^2=0\)
\(x^2=4\)
\(x^2=8\)
So then the roots are
\(x=0\)
\(x=-2\)
\(x=2\)
\(x=-\sqrt{8}\)
\(x=\sqrt{8}\)
5 roots. Answer: E
Note, since we are not asked to find the actual roots, we only need to determine the number of roots. After the first step of factoring out the \(x^2\), we have \(x^2\) and a 4th order expression. A 4th order expression will have 4 roots (unless it's a perfect square, but our expression is not a perfect square). So the total number of roots will be 4 from the 4th order expression, and one from the \(x^2\). Total roots = 5.
21 Mar 2017, 19:59
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Hi mesutthefail,
The GMAT often tests you on rules/patterns that you know, but sometimes in ways that you're not used to thinking about. This prompt is really just about factoring and Classic Quadratics, but it looks a lot more complicated than it actually is.
A big part of properly dealing with a Quant question on the GMAT is in how you organize the information and 'simplify' what you've been given. This prompt starts us off with....
X^6 –12X^4 + 32X^2 = 0
This certainly looks complex, but if you think about how you can simplify it, then you'll recognize that can 'factor out' X^2 from each term. This gives us...
(X^2)(X^4 - 12X^2 + 32) = 0
Now we have something a bit more manageable. While you're probably used to thinking of Quadratics such as X^2 + 6X + 5 as (X+1)(X+5), that same pattern exists here - it's just the exponents are slightly different (even though the math rules are exactly the SAME). We can further rewrite the above equation as...
(X^2)(X^2 -4)(X^2 - 8) = 0
At this point, you don't really need to calculate much, since each 'piece' of the product should remind you of a pattern that you already know.....
X^2 = 0 --> 1 solution
(X^2 - 4) = 0 --> 2 solutions
(X^2 - 8) = 0 --> 2 solutions
Final Answer:
GMAT assassins aren't born, they're made,
Rich
The GMAT often tests you on rules/patterns that you know, but sometimes in ways that you're not used to thinking about. This prompt is really just about factoring and Classic Quadratics, but it looks a lot more complicated than it actually is.
A big part of properly dealing with a Quant question on the GMAT is in how you organize the information and 'simplify' what you've been given. This prompt starts us off with....
X^6 –12X^4 + 32X^2 = 0
This certainly looks complex, but if you think about how you can simplify it, then you'll recognize that can 'factor out' X^2 from each term. This gives us...
(X^2)(X^4 - 12X^2 + 32) = 0
Now we have something a bit more manageable. While you're probably used to thinking of Quadratics such as X^2 + 6X + 5 as (X+1)(X+5), that same pattern exists here - it's just the exponents are slightly different (even though the math rules are exactly the SAME). We can further rewrite the above equation as...
(X^2)(X^2 -4)(X^2 - 8) = 0
At this point, you don't really need to calculate much, since each 'piece' of the product should remind you of a pattern that you already know.....
X^2 = 0 --> 1 solution
(X^2 - 4) = 0 --> 2 solutions
(X^2 - 8) = 0 --> 2 solutions
Final Answer:
GMAT assassins aren't born, they're made,
Rich
