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February 15th, 2018, 06:51 AM   #1
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[ASK] About Polynomial

A polynomial f(x) = $\displaystyle 2x^3-5x^2+ax+18$ is divisible by (x - 3). The result of that polynomial f(x) divided by (x - 1) is ....
A. $\displaystyle 2x^2-7x+2$
B. $\displaystyle 2x^2+7x-2$
C. $\displaystyle 2x^2-7x-2$
D. $\displaystyle x^2-6x-2$
E. $\displaystyle x^2-6x+3$

I got a + 3 = -6 and so a = -9 and f(x) = $\displaystyle 2x^3-5x^2-9x+18$, but when I divided it with x - 1 I got $\displaystyle x^2-3x+6$ with the remainder 24. I'm quite sure that the question isn't wrong since it's in a national level exam, but where did I go wrong?
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February 15th, 2018, 07:06 AM   #2
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Must be an error in the question somewhere ...

$f(3) = 0 \implies a = -9 \implies f(1) = 6 \implies f(x)$ is not divisible by $x-1$.
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February 15th, 2018, 07:24 AM   #3
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If the leading coefficient of both divisor and dividend are not equal, the result cannot have a leading coefficient of 1. There's clearly a problem with your division.
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February 15th, 2018, 12:19 PM   #4
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Long division:
(2x^3 - 5x^2 + ax + 18 ) / (x - 3)
= 2x^2 + x + 3 with remainder = 27

Last edited by Denis; February 15th, 2018 at 12:42 PM.
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February 15th, 2018, 02:03 PM   #5
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I suspect that the question should be asking for the result of the polynomial $f(x)$ divided by $(x + 1)$ instead of $(x - 1)$.

Dividing $2x^3 - 5x^2 - 9x + 18$ by $(x + 1)$ gives $2x^2 - 7x - 2$ (with remainder 20), which corresponds to answer choice C.
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February 15th, 2018, 05:29 PM   #6
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Quote:
Originally Posted by skipjack View Post
I suspect that the question should be asking for the result of the polynomial $f(x)$ divided by $(x + 1)$ instead of $(x - 1)$.
That's a possible reason since the paper was photocopied and sometimes the vertical line on the plus sign doesn't show up thus making people think it is a minus sign.
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