Question

how to solve a cubic equation? Please explain in a simple manner because I have trouble in understanding them and what is this Synthetic division????? How is it helpful??

Answer 1

here's a general concept..find a root then use synthetic division to divide the given polynomial by that root...the quotient will give you a quadratic equation,.you can use quadratic formula to find the remaining two roots

Answer 2

now...going more specifically..synthetic division is a way of dividing polynomials

Answer 3

to solve a cubic equation there are lot of methods... here you can find details http://en.wikipedia.org/wiki/Cubic_function

Answer 4

wouldn't quadratic formula be easy

Answer 5

well quadratic only suits when it can be factored easily...like in case X^3-7X+9=0 we can't apply quadratic

Answer 6

Please explain step by step the process of finding the root of the cubic polynomial which of the easiest way

Answer 7

Actually I would say..... to use FACTOR theorem and Remainder theorem is better

Answer 8

actually...remainder/factor theorem + rational roots theorem...are you familiar with either terms?

Answer 9

I remember using them but its been a long gap so please explain again

Answer 10

the truth is that there is no easy way

Answer 11

if the polynomial has been cooked up to make it easy, that is, if it has been specifically constructed to have integer or rational roots, then you can find them using the rational root theorem. but if you just write down some random third degree polynomial, you will have to use a long and ugly formula to find the roots. i believe it is in the wiki link above

Answer 12

one way, if given the form:

x^3 + ax^2 + bx + c = n is to eliminate the x^2 by a suitable replacement of:
x = y-a/3

this will get you the form: y^3 + py = q; which somehow is simpler to deal with :/

Answer 13

I will prefer you the way of : factor theorem

Answer 14

\[\large{x^3+ax^2+bx+c}\]
find the factors of c :\(\large{\textbf{plus and minus factors are included}}\)
then by common sense and trial and error find only 1 zero of the given polynomial

let us suppose we have the polynomial p(x) and a zero of this polynomial (a)

so (x-a) will be the factor of p(x)

(x-a)(b)=p(x)
where b is another factor of p(x) , b may include 2 factors or more..

divide p(x) by (x-a) by long division or any other method

you will get a polynomial suppose it is (x-b)

hence we have (x-a)(x-b) =p(x)

that is (x-a) and (x-b) are the factors of p(x)

Answer 15

as satellite has already pointed out, that will only work if the poly has been constructed to conform to that method.

rational roots test is effective only if there exists rational roots to the poly.

Answer 16

you can do trial and error tests with consecutive integers (a < b) such that:

f(a) < 0 and f(b)>0 this at least lets you know a root exists somewhere in the interval (a,b)

Answer 17

That is a good point though :)

Answer 18

can you guys teach me this with an example please..it would be helpful for me very much..please

Answer 19

see you can find the first root by hit and trial
and rest two roots by horners method or factorisation

Answer 20

eg:
\[x ^{3}-7x+6=0\]

Answer 21

are you there?????

Answer 22

what next @nitz

Answer 23

so first root of this equation is 1 as (1)^3-7(1)+6=-6+6=0
thus it satisfies the equation ......
so by hit and trial .the first root is 1
right?????

Answer 24

basically in hit and trial,we substitute values -1,0,1,.. and check out which satisfies the equation....generally you need to check just -1,1,2,3,4.....just starting nos.

Answer 25

ok got it then

Answer 26

no actually i was trying to figure out how to show you synthetic division here.........

Answer 27

hmm can you please draw it

Answer 28

http://www.mash.dept.shef.ac.uk/Resources/web-cubicequations-john.pdf

Answer 29

can you please check this link...here is another example

Answer 30

its not possible to draw...if u have any problem i can tell
sorry

Answer 31

what to draw i can..

Answer 32

synthetic division method...

Answer 33

what is the dividend and divisor @nitz ?

Answer 34

the equation is above x^3-7x+6=0

Answer 35

one root is 1

Answer 36

oh k so this means

x-1 is a factor of x^3-7x+6

Answer 37

got it guys thanks @nitz