Question 61715: Can you help me please
Find all three zeros of f(x)=x^3-4X^2+13x+50 given that 3+4i is a zero
thank you
Found 2 solutions by ankor@dixie-net.com, jai_kos:
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! Find all three zeros of f(x)=x^3-4X^2+13x+50 given that 3+4i is a zero
:
Find the expression for 3 + 4i, reverse the procedure of completing the square:
x = 3 + 4i
x - 3 = 4i
:
Square both sides
(x-3)^2 = 4i * 4i
:
x^2 - 6x + 9 = 16(i^2)
:
Remember i^2 = -1
x^2 - 6x + 9 = 16(-1)
x^2 - 6x + 9 = -16
x^2 - 6x + 9 + 16 = 0
x^2 - 6x + 25 = 0
:
We can see that the last term is 50 so the above expression has to multiplied by the factor (x + 2) to get x^3 - 4X^2 + 13x + 50
:
The 3 zeros: -2, 3+4i, 3-4i
Answer by jai_kos(139)  (Show Source):
You can put this solution on YOUR website! F(x) = x^3 - 4x^2 + 13 x + 50 ---->(1)
Check the last number, which is 50
Now find the factors of 50,
That is 50 ---- 1 , 2 , 5 ,25
So take all the numbers ---- 1 , -1 ,2 ,-2 ,5,-5,25 , -25
Now take the first factor 1. put in equation(1), we get
F(1) = 1^3 - 4*1^2 + 13*1+ 50 = 48
F(1) is not equal to zero.
Like this find for different numbers,
Then we find that F(-2) is equal to zero.
So x = -2 is a zero. ------>(2)
Now rewrite the above equation we find that...
x+2 = 0
Now take this and divide equation (1),
x^2 - 6x + 25
---------------------------
(x+2) | x^3 - 4x^2 + 13 x + 50 Subtract, we
get
| x^3 + 2x^2
| - 6x^2 +13x
| - 6x^2 - 12x Subtract , we get
| 25x + 50
25x + 50 Subtract, we get
0
We get x^2 - 6x + 25
Simplify this quadratic equation, we get
x = - (-6) +- sqrt(6^2 - 4* 1* 25
2
x = 3 + 4i and x = 3 - 4i
Therefore the zeroes are x = 2 and x = 3 + 4i , x = 3 - 4i
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