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# Math Question

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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
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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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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
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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
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ok. YW!

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

Jan 12, 2019
#8
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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

Jan 12, 2019
edited by CPhill  Jan 12, 2019
#10
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it was -20... i got every case except for -7

Jan 12, 2019
#11
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5 is extraneous, but that makes -19, so you missed another case, i dont know which

Jan 12, 2019
#12
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wait.. you are right -5 does not work :P

oops

Jan 12, 2019