SOLUTION: If (2x-1) is a factor of f(x)=2x^3-5x^2-kx+3 find the value of k. Find the other two factors of f(x)=0

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Question 366845: If (2x-1) is a factor of f(x)=2x^3-5x^2-kx+3 find the value of k.
Find the other two factors of f(x)=0
Answer by Jk22(389) (Show Source):
You can put this solution on YOUR website!
Dividing the polynomials gives :

2x^3 - 5x^2 - kx + 3 | 2x - 1
------------------------------
2x^3 - x^2............| x^2 - 2x - (k+2)/2
----------------------|
-4x^2 - kx + 3........|
-4x^2 + 2x............|
----------------------|
-(k+2)x + 3...........|
-(k+2)x + (k+2)/2.....|
----------------------|
3 - (k+2)/2

the remainder should equals zero, hence k+2 = 6, k = 4

we can check another way by the fact the 1/2 has to be a solution :

2/8 - 5/4 - 2 + 3 = 1/4 - 5/4 + 1 = -4/4 + 1 = 0

the remaining polynomial is : x^2 - 2x - 3 = (x-3)(x+1) by Viete's formula.

the 2 other roots are : 3 and -1

check : for x=3 : 2*27 - 5*9 - 4*3 + 3 = 54 - 45 - 12 + 3 = 9 - 12 + 3 = 0

x=-1 : -2 - 5 + 4 + 3 = -7 + 7 = 0