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Difference between revisions of "2010 AIME II Problems/Problem 7"

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Since <math>a,b,c\in{R}</math>, the imaginary part of a,b,c must be 0.
 
Since <math>a,b,c\in{R}</math>, the imaginary part of a,b,c must be 0.
  
Start with a, since it's the easiest one to do: <math>y+3+y+9+2y=0, y=-3</math>
+
Start with a, since it's the easiest one to do: <math>y+3+y+9+2y=0, y=-3</math>,
  
and therefore: <math>x_1 = x</math>, <math>x_2 = x+6i</math>, <math>x_3 = 2x-4-6i</math>
+
and therefore: <math>x_1 = x</math>, <math>x_2 = x+6i</math>, <math>x_3 = 2x-4-6i</math>.
  
 
Now, do the part where the imaginary part of c is 0, since it's the second easiest one to do:  
 
Now, do the part where the imaginary part of c is 0, since it's the second easiest one to do:  
<math>x(x+6i)(2x-4-6i)</math>, the imaginary part is: <math>6x^2-24x</math>, which is 0, and therefore x=4, since x=0 doesn't work.
+
<math>x(x+6i)(2x-4-6i)</math>. The imaginary part is: <math>6x^2-24x</math>, which is 0, and therefore x=4, since x=0 doesn't work.
  
So now, <math>x_1 = 4, x_2 = 4+6i, x_3 = 4-6i</math>
+
So now, <math>x_1 = 4, x_2 = 4+6i, x_3 = 4-6i</math>,
  
and therefore: <math>a=-12, b=84, c=-208</math>, and finally, we have <math>|a+b+c|=|-12+84-208|=\boxed{136}</math>.
+
and therefore: <math>a=-12, b=84, c=-208</math>. Finally, we have <math>|a+b+c|=|-12+84-208|=\boxed{136}</math>.
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2010|num-b=6|num-a=8|n=II}}
 
{{AIME box|year=2010|num-b=6|num-a=8|n=II}}

Revision as of 21:56, 5 March 2012

Problem 7

Let $P(z)=x^3+ax^2+bx+c$, where a, b, and c are real. There exists a complex number $w$ such that the three roots of $P(z)$ are $w+3i$, $w+9i$, and $2w-4$, where $i^2=-1$. Find $|a+b+c|$.

Solution

Set $w=x+yi$, so $x_1 = x+(y+3)i$, $x_2 = x+(y+9)i$, $x_3 = 2x-4+2yi$.

Since $a,b,c\in{R}$, the imaginary part of a,b,c must be 0.

Start with a, since it's the easiest one to do: $y+3+y+9+2y=0, y=-3$,

and therefore: $x_1 = x$, $x_2 = x+6i$, $x_3 = 2x-4-6i$.

Now, do the part where the imaginary part of c is 0, since it's the second easiest one to do: $x(x+6i)(2x-4-6i)$. The imaginary part is: $6x^2-24x$, which is 0, and therefore x=4, since x=0 doesn't work.

So now, $x_1 = 4, x_2 = 4+6i, x_3 = 4-6i$,

and therefore: $a=-12, b=84, c=-208$. Finally, we have $|a+b+c|=|-12+84-208|=\boxed{136}$.

See also

2010 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 6
Followed by
Problem 8
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All AIME Problems and Solutions
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