Question 27026: I need to solve z^3 - 10z^2 -32=0
I tried factors +/_ 1,2,4,8,16,32 but can't solve the values for the rational zeros.
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! YOUR PROBLEM IS EXACTLY SIMILAR TO THE FOLLOWING EXAMPLE.SAME COMMENTS APPLY.COME BACK IF YOU WANT AFTER READING THE FOLLOWING.
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I have tried +/- 1,2,3,4,6,12 to solve
x^4 -4x^3 -x^2-16x-12=0 but can't find the rational zeros.
YOU ARE VERY CORRECT..POOR SOUL!YOU MUST HAVE SPENT A LONG TIME...YOU DESERVE COMPLEMENTS FOR YOUR SINCERE ATTEMPT...FIRSTLY THIS PROBLEM SHOULD NOT BE GIVEN TO YOU,THOUGH, I DONT KNOW YOUR COURSE OF STUDY.HAVING SAID THAT,IN CASE IT BECOMES NECESSARY,LET ME GIVE YOU A COUPLE OF IDEAS TO PROCEED IN SUCH CASES...
TRY TO FACTORISE THE 4TH. DEGREE EQN. INTO 2 QUADRATIC FACTORS.USE PROPERTY OF IDENTITIES ,METHOD OF UNDETERMINATE COEFFICIENTS AND A LITTLE GUESS TO SIMPLIFY WORK.
THEN SOLVE EACH QUADRATIC BY THE STANDARD FORMULA.
SO WE WRITE THE EQN. AS
x^4 -4x^3 -x^2-16x-12 = (X^2+AX+4)(X^2+BX-3)..WE USED THE FACT THAT COEFFICIENT OF X^4 IS 1 WHICH IS OK AS THERE IS NO OTHER POSSIBILITY.BUT CONSTANT TERM OF
-12 CAN BE SPLIT IN DIFFERENT WAYS..1 AND 12 ,2 AND 6,3 AND 4 ETC..ANY WAY TO ILLUSTRATE THE METHOD , I HAVE JUST TAKEN IT AS 3 AND 4.IT DOES NOT MATTER WHETHER WE PUT +3 OR -3,SINCE A AND B AS UNKNOWNS WILL TAKE CARE OF THAT .SO
x^4 -4x^3 -x^2-16x-12 = X^4+X^3(A+B)+X^2(-3+4-3A+4B)+X(-3A+4B)-12
A+B=-4.............I
1-3A+4B= -1........II
-3A+4B=-16...........III
EQNI*3+EQN.III
7B=-28
B=-4
SUBSTITUTING IN EQN.I , WE GET
A=0
BUT A=0 AND B=-4 DOES NOT SATISFY EQN.II...SO THIS IS NOT A CORRECT SPLIT UP FOR -12..MAY BE WE SHOUD TRY 1&12 OR 2&6 ETC..
BUT AS THIS ONLY AN ILLUSTRATION OF THE METHOD LET US CORRECT THE PROBLEM BY CHANGING X^2 COEFFICIEN TO +1 INSTEAD OF -1..THAT IS THE PROBLEM IS
x^4 -4x^3 +x^2-16x-12 =0..THEN A=0 AND B=-4 ARE OK SO WE GET
x^4 -4x^3 +x^2-16x-12 =0=(X^2+4)(X^2-4X-3)....
SO X^+4=0..OR..X==2I AND -2I
X^2-4X-3=0
SO 
THAT COMPLETES THE SOLUTION WHERE YOU GET 2 IRRATIONAL ROOTS AND 2 IMAGINARY ROOTS..BUT AS I SAID BEFORE..THIS PROBLEM SHOULD NOT BE GIVEN TO YOU IF YOU ARE IN SCHOOL LEVEL.
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