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Consider f(x) = x^3 - 2x^2 - 5x + k Find the value of k if (x+2) is a factor of f(x)...

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abctest123 | Student, Undergraduate | eNotes Newbie

Posted April 10, 2010 at 4:50 PM via web

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Consider f(x) = x^3 - 2x^2 - 5x + k

Find the value of k if (x+2) is a factor of f(x)

 

I'm not sure if this is the answer because it says this question is worth 6 marks:

 

f(-2) = (-2)^3 -2(-2)^2 -5(-2) +k = 0
= -8 -8 +10 + k
= -16+10 +k

k = 6

 

????

2 Answers | Add Yours

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neela | High School Teacher | Valedictorian

Posted April 10, 2010 at 5:34 PM (Answer #1)

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To find the value of k if f(x) =  x^3 - 2x^2 - 5x + k is divisible by (x+2).

Solution:

If  x^3 - 2x^2 - 5x + k is divided by x+2, the n you are getting a n expression, Q(x) as quotient and a remainder, R. So

f(x) /(x+2) = Q(x)+ R/(x+2). Or

f(x) = (x+2) Q(x)+R. Putting x = -2, we get:

f(-2) =(-2)^3-2(-2)^2-5(-2)+k = (-2+2)*Q(-2)+R.

f(-2) = -8-8+10+k = 0+R.  But R should be zero as x+2 divides f(x) without remainder. Therefore,

f(-2) =-6+k=0. Or

k=6.

Note: What you have done is perfectly alright. Only that you could have added a little explanation.

 

 

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krishna-agrawala | College Teacher | Valedictorian

Posted April 10, 2010 at 9:53 PM (Answer #2)

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The solution given in the question itself is definitely valid is correct.

As (x + 2) is one of the factors of f(x), the value of (x + 2) when x is equal to -2 is zero.

Therefor, when x = -2, we can equate f(-2) to 0 and then solve the equation for value of x, as has been done. However there are other ways of solving this question also. One such way is given below.

Given:

f(x) = x^3 - 2x^2 - 5x - k

and that (x + 2) is a factor of f(x)

Therefore if we divide f(x) bu (x + 2) the remainder should be 0.

Dividing f(x) by (x + 2) we get:

f(x)/((x + 2) = (x^3 - 2x^2 - 5x - k)/(x + 2)

= (x^2 - 4x + 3) + (k - 6)/(x + 2)

As k-6 in the above expression is a constant, it is the remainder. This remainder has to be 0.

Therefore:

k - 6 = 0

Or:

k = 6

Answer:

k = 6

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