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by oddball » Fri Oct 29, 2010 6:08 am
When (x^3 - 2*x^2 + 2*k*x + 4) is divided by (x - 1), the remainder is 5. What is the value of k?
  • (A) 0
    (B) 1
    (C) 2
    (D) 3
    (E) 4
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by shovan85 » Fri Oct 29, 2010 6:37 am
oddball wrote:When (x^3 - 2*x^2 + 2*k*x + 4) is divided by (x - 1), the remainder is 5. What is the value of k?
  • (A) 0
    (B) 1
    (C) 2
    (D) 3
    (E) 4
IMO B

You know general process of division. Just follow that. Here the Divisor is (x-1). Your objective should be to remove the x till the end so that you will get a remainder which will be independent of x. Then Equate that remainder to 5 and get the answer. See the Image below I have listed it step wise.
Hope this is correct and this helps.
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by GMATGuruNY » Fri Oct 29, 2010 8:39 am
oddball wrote:When (x^3 - 2*x^2 + 2*k*x + 4) is divided by (x - 1), the remainder is 5. What is the value of k?
  • (A) 0
    (B) 1
    (C) 2
    (D) 3
    (E) 4
Easiest and safest approach would be to plug in. In questions that involve exponents, 10 is a good number to plug in because it can be easily raised to different powers.

Plug in x=10.

x^3 - 2*x^2 + 2*k*x + 4 = 10^3 - 2(10^2) + 2k*10 + 4 = 804 + 20k
x-1 = 10-1 = 9

Not let's plug in the answer choices for k. When we plug our value for k into 804 + 20k, we need to divide by 9 to see which answer choice yields a remainder of 5.

Answer choice C: k=2
804 + 20(2) = 844
844/9 = 97 R7
Doesn't work.

Answer choice B: k=1
804 + 20(1) = 824
824/9 = 91 R5
Success!

The correct answer is B.
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by Rahul@gurome » Fri Oct 29, 2010 8:53 am
oddball wrote:When (x^3 - 2*x^2 + 2*k*x + 4) is divided by (x - 1), the remainder is 5. What is the value of k?
  • (A) 0
    (B) 1
    (C) 2
    (D) 3
    (E) 4
This kind of problems can be solved with minimum calculation if we apply the polynomial remainder theorem. The theorem states that, if a polynomial p(x) is divided by (x - a) then the remainder is nothing but p(a). Let's see, how it is possible!

Say, p(x) = (x - a)*q(x) + r
Thus, when p(x) is divided by (x - a), the quotient is q(x) and remainder is r (which doesn't contain x).
Now, if we set x = a, p(a) = (a - a)*q(a) + r => r = p(a)

For more detailed discussion: https://www.purplemath.com/modules/remaindr.htm

Now, for this question: p(x) = (x^3 - 2*x^2 + 2*k*x + 4) => p(1) = (1 - 2 + 2k + 4) = (3 + 4k)
Therefore, (3 + 4k) = 5 => k = 1.

The correct answer is B.
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