QUESTIONS - POLYNOMIALS - COMPLEX ROOTS

QUESTIONS - SUNNY - POLYNOMIALS

QUESTION BY SUNNY ON POLYNOMIALS - COMPLEX ROOTS
FIND ALL ROOTS OF THE POLYNOMIAL-REAL AND COMPLEX
6X^5+13X^4-8X^3+16X^2-14X+3 = 0
IF FACTORS OF CONSTANT ARE C1,C2,C3 ETC…AND
FACTORS OF HIGHEST DEGREE TERM ARE H1,H2,H3 ETC..
THEN POSSIBLE RATIONAL ROOTS ARE + OR - C1/H1,C1/H2….C2/H1,C2/H2..ETC.
SO TRY RATIONAL ROOTS FIRST
6X^5+13X^4-8X^3+16X^2-14X+3
FACTORS OF CONSTANT TERM 3 ARE 1 AND 3.
FACTORS OF COEFFICIENT OF X^4 TERM - 6 - ARE 1,2,3 AND 6.
SO POSSIBLE RATIONAL ROOTS ARE + OR - 1,3,1/2,1/3,ETC
BY TRIAL WE FIND X= -3 IS A ROOT SINCE
6*(-3)^5+13*(-3)^4-8*(-3)^3+16*(-3)^2-14*(-3)+3 = 0
SO X =-3 THAT IS X+3=0 AND X+3 IS A FACTOR
DIVIDING WITH X+3
-3) 6 +13 - 8 + 16 -14 +3
0 -18 +15 -21 +15 -3
-----------------------------------
6 -5 +7 - 5 + 1 +0 REMAINDER
QUOTIENT = 6X^4 - 5X^3 + 7X^2 -5X +1

BY TRIAL WE FIND 1/2 AND 1/3 ARE ROOTS SINCE F(1/2) AND F(1/3) EQUAL ZERO
SO X=1/2 THAT IS 2X-1 = 0 AND HENCE 2X-1 IS A FACTOR.
SIMILARLY X=1/3 THAT IS 3X-1 = 0 AND HENCE 3X-1 IS A FACTOR.
DIVIDING SUCCESSIVELY WITH THESE 2 FACTORS

2X-1) 6X^4 - 5X^3 + 7X^2 -5X + 1 (3X^3 -X^2 +3X -1 =QUOTIENT
6X^4 -3X^3
-----------------------------------
-2X^3+7X^2
-2X^3+X^2
-----------------------------
6X^2 - 5X
6X^2 - 3X
-----------------
-2X +1
-2X +1
-----------------------
0

3X-1) 3X^3 - X^2 +3X -1 (X^2 + 1 = QUOTIENT
3X^3 - X^2
--------------------------
3X - 1
3X - 1
----------------------------
0

X^2+1 = (X-I) (X+I)…..THAT X = I AND -I ARE ROOTS.
SO ALL ROOTS ARE X = -3 , 1/2 , 1/3 ,I AND -I
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