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The process of finding complex zeros of polynomial functions using the given information and the fundamental theorem of algebra. It includes various examples with detailed steps and explanations.
Typology: Lecture notes
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= (^) ( x^2 − 8 x + (^16) ) (( x − 3) − 2 i ) (( x − 3) + 2 i ) = (^) ( x^2 − 8 x + (^16) ) ( x^2 − 6 x + 9 − 4 i^2 ) = (^) ( x^2 − 8 x + (^16) ) ( x^2 − 6 x + (^13) ) = x^4 − 6 x^3 + 13 x^2 − 8 x^3 + 48 x^2 − 104 x + 16 x^2 − 96 x + 208 = x^4 − 14 x^3 + 77 x^2 − 200 x + 208
= (^) ( x^2 − i^2 ) ( x^2 − 2 x + 1 − 4 i^2 ) = (^) ( x^2 + (^1) ) ( x^2 − 2 x + (^5) ) = x^4 − 2 x^3 + 5 x^2 + 1 x^2 − 2 x + 5 = x^4 − 2 x^3 + 6 x^2 − 2 x + 5
= ( x − 2) (^) ( x^2 − i^2 ) (( x − 1) − i )( ( x − 1)+ i ) = ( x − 2) (^) ( x^2 + (^1) ) ( x^2 − 2 x + 1 − i^2 ) = (^) ( x^3 − 2 x^2 + x − (^2) )( x^2 − 2 x + (^2) ) = x^5 − 2 x^4 + 2 x^3 − 2 x^4 + 4 x^3 − 4 x^2 + x^3 − 2 x^2 + 2 x − 2 x^2 + 4 x − 4 = x^5 − 4 x^4 + 7 x^3 − 8 x^2 + 6 x − 4
= (^) ( x^2 − i^2 ) (( x − 4) − i ) (( x − 4) + i ) (( x − 2) − i ) (( x − 2 )+ i ) = (^) ( x^2 + (^1) ) ( x^2 − 8 x + 16 − i^2 ) ( x^2 − 4 x + 4 − i^2 ) = (^) ( x^2 + (^1) )( x^2 − 8 x + (^17) ) ( x^2 − 4 x + (^5) ) = (^) ( x^4 − 8 x^3 + 17 x^2 + x^2 − 8 x + (^17) ) ( x^2 − 4 x + (^5) ) = (^) ( x^4 − 8 x^3 + 18 x^2 − 8 x + (^17) )( x^2 − 4 x + (^5) ) = x^6 − 4 x^5 + 5 x^4 − 8 x^5 + 32 x^4 − 40 x^3 + 18 x^4 − 72 x^3 + 90 x^2 − 8 x^3
Thus, ( x − 2 i )( x + 2 i ) = x^2 + 4 is a factor of f. Using division to find the other factor:
x^2 + 4 2 x^4 + 5 x^3 + 5 x^2 + 20 x − 12 2 x^4 + 8 x^2 5 x^3 − 3 x^2 + 20 x 5 x^3 + 20 x − 3 x^2 − 12 − 3 x^2 − 12
2 x^2 + 5 x − 3 )
2 x^2 + 5 x − 3 = ( 2 x − 1)( x + 3) are factors and the remaining zeros are 12 and − 3. The zeros of f are 2 i , − 2 i , − 3 , 12.
x^2 + 9 3 x^4 + 5 x^3 + 25 x^2 + 45 x − 18 3 x^4 + 27 x^2 5 x^3 − 2 x^2 + 45 x 5 x^3 + 45 x − 2 x^2 − 18 − 2 x^2 − 18
3 x^2 + 5 x − 2 )
3 x^2 + 5 x − 2 = (3 x − 1)( x + 2) are factors and the remaining zeros are 13 and − 2. The zeros of h are 3 i , − 3 i , − 2, 13.
x^2 − 6 x + 13 x^4 − 9 x^3 + 21 x^2 + 21 x − 130 x^4 − 6 x^3 + 13 x^2 − 3 x^3 + 8 x^2 + 21 x − 3 x^3 + 18 x^2 − 39 x − 10 x^2 + 60 x − 130 − 10 x^2 + 60 x − 130
x^2 − 3 x − 10 )
x^2 − 3 x − 10 = ( x + 2)( x − 5) are factors and the remaining zeros are –2 and 5. The zeros of h are 3 − 2 i , 3 + 2 i , − 2, 5.
x^2 − 2 x + 10 x^4 − 7 x^3 + 14 x^2 − 38 x − 60 x^4 − 2 x^3 + 10 x^2 − 5 x^3 + 4 x^2 − 38 x − 5 x^3 + 10 x^2 − 50 x − 6 x^2 + 12 x − 60 − 6 x^2 + 12 x − 60
x^2 − 5 x − 6 )
x^2 − 5 x − 6 = ( x + 1)( x − 6 ) are factors and the remaining zeros are –1 and 6. The zeros of f are 1 + 3 i , 1− 3 i , − 1 , 6.
Thus, ( x − 4 i )( x + 4 i ) = x^2 + 16 is a factor of h. Using division to find the other factor:
x^2 + 16 3 x^5 + 2 x^4 + 15 x^3 + 10 x^2 − 528 x − 352 3 x^5 + 48 x^3 2 x^4 − 33 x^3 + 10 x^2 2 x^4 + 32 x^2 − 33 x^3 − 22 x^2 − 528 x − 33 x^3 − 528 x − 22 x^2 − 352 − 22 x^2 − 352
3 x^3 + 2 x^2 − 33 x − 22 )
3 x^3 + 2 x^2 − 33 x − 22 = x^2 (3 x + 2) − 11(3 x + 2) = ( 3 x + 2 ) ( x^2 − 11) = (3 x + 2 )( x − (^11) ) ( x + (^11) ) are factors and the remaining zeros are − 32 , 11, and − 11. The zeros of h are 4 i , − 4 i , − 11, 11, − 32.
Using the quadratic formula to find the zeros of x^2 − 6 x + 13 = 0 :
x =
6 ± 4 i 2 =^3 ±^2 i^. The complex zeros are 2, 3 − 2 i , 3 + 2 i.
p q = ±^1 ,^ ±^5 ,^ ±^ 17,^ ±^85 Step 4: Using synthetic division: −5 1 13 57 85 − 5 − 40 − 85 1 8 17 0
)
Since the remainder is 0, x + 5 is a factor. The other factor is the quotient: x^2 + 8 x + 17.
Using the quadratic formula to find the zeros of x^2 + 8 x + 17 = 0 :
x =
− 8 ± 2 i 2 = −^4 ±^ i^. The complex zeros are − 5 , − 4 − i , − 4 + i.
The zeros are: − 2 i , − i , i , 2 i.
The zeros are: − 3 i , − 2 i , 2 i , 3 i.
Step 4: Using synthetic division: −3 1 2 22 50 − 75 − 3 3 − 75 75 1 − 1 25 − 25 0
)
Since the remainder is 0, x + 3 is a factor. The other factor is the quotient: x^3 − x^2 + 25 x − 25 = x^2 ( x − 1) + 25( x − 1) = ( x − 1)( x^2 + (^25) ) = ( x − 1)( x + 5 i )( x − 5 i )
The complex zeros are − 3 , 1 , − 5 i , 5 i.
)
Since the remainder is 0, x + 7 is a factor. The other factor is the quotient: x^3 − 4 x^2 + 9 x − 36 = x^2 ( x − 4 )+ 9( x − 4 ) = ( x − 4 )( x^2 + (^9) ) = ( x − 4 ) ( x + 3 i )( x − 3 i )
The complex zeros are −7, 4, − 3 i , 3 i.
p = ± 1 , ± 2, ± 4, ± 13, ± 26, ± 52; q = ± 1 , ± 3; p q