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The sum of all integral values of \(x\) for which \(\frac{18({x}^{2}-7x+10)}{{x}^{3}-6{x}^{2}+3x+10}\) has an integral value is...

 

Thanks in advance for the help everyone!

 Jan 12, 2019
 #1
avatar+533 
+1

Use synthetic division twice on the two polynomials to get the whole equation as 18/(x+1).

 

the roots of the denominator of the original cubic are -1, 2, and 5 so those will not work. the remaining ones are the the positive and negative factors of 18, subtracted by 1 so you get -19, -10, -5, -3, -2, -1(extraneous), 0, 1, 2(extraneous), 5(extraneous), 8, and 17. adding up the ones that work, you get 

 

-19 - 10 - 5 - 3 - 2 + 0 + 1 + 8 + 17. with a little mental math, you get -13.

 

 

HOPE THIS HELPED!

 Jan 12, 2019
 #2
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+1

Thanks for the help!

 

But I have some questions. What is synthetic division? And what do you mean by "the roots of the denominator of the original cubic are -1, 2, and 5". I think I know what you mean by that but the wording is off. 

 

I'm not as good at math as you lol

 Jan 12, 2019
 #3
avatar+533 
0

i mean x^3-6x^2+3x+10

 

and synthetic division is just a quicker way of doing long division, its almost the same thing

 Jan 12, 2019
 #4
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Ok thanks, I check it out and ask you if I have anymore questions

Guest Jan 12, 2019
 #6
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Wait -13, isn't one of the answers in my book. The options there are -17, -19, -12, -14 and -20. Thanks!

Guest Jan 12, 2019
 #5
avatar+533 
0

ok. YW!

 Jan 12, 2019
 #7
avatar+533 
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can you check my work? i think i got every case though..

 Jan 12, 2019
 #8
avatar+102947 
+1

Here's my best attempt....

 

The top factors as 18 (x - 5) ( x - 2)

The bottom factors as  (x + 1) ( x - 5) ( x - 2)

 

So  we are left with    18 / (x + 1)

 

The divisors of 18 are   ± (1, 2, 3, 6, 9, 18)

 

This means that we have integer values when  x =   -2 , -3  -4, -7, -10, -19, 0, 1, 5, 8, 17 

 

Note that we cannot use x =   -1   or x = 2 because they make an original denominator = 0

 

So  the sum of these integer values are  - 45 + 31  =   - 14

 

 

cool cool cool

 Jan 12, 2019
edited by CPhill  Jan 12, 2019
 #10
avatar+533 
+2

it was -20... i got every case except for -7

 Jan 12, 2019
 #11
avatar+533 
+1

5 is extraneous, but that makes -19, so you missed another case, i dont know which 

 Jan 12, 2019
 #12
avatar+533 
+1

wait.. you are right -5 does not work :P

oops

 Jan 12, 2019

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