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Theory of Equations..
by Total Gadha - Saturday, 3 March 2007, 03:06 PM
 

GENERAL EQUATION OF Nth DEGREE

Let polynomial f(x) = a0xn + a1xn - 1 + a2xn - 2 + ... + an. where a0, a1, a2, ..an are rational numbers and n > 0. Then the values of x for which f(x) reduces to zero are called root of the equation f(x) = 0. The highest whole number power of x is called the degree of the equation.

For example

x4 - 3x3 + 4x2 + x + 1 = 0 is an equation with degree four.

x5 - 6x4 + 3x2 + 1 = 0 is an equation with degree five.

ax + b = 0 is called the linear equation.

ax2 + bx + c = 0 is called the quadratic equation.

ax3 + bx2 + cx + d = 0 is called the cubic equation.

Properties of equations and their roots

  • Every equation of the nth degree has exactly n roots.

For example, the equation x3 + 4x2 + 1 = 0 has 3 roots,

The equation x5 - x + 2 = 0 has 5 roots, and so on…

   nature of roots

  • In an equation with real coefficients imaginary roots occur in pairs i.e. if a + ib is a root of the equation f(x) = 0, then a - ib will also be a root of the same equation. For example, if 2 + 3i is a root of equation f(x) = 0, 2 - 3i is also a root.

    surd roots occur in pairs

  • If the coefficients of an equation are all positive then the equation has no positive root. Hence, the equation 2x4 + 3x2 + 5x + 1 = 0 has no positive root.
  • If the coefficients of even powers of x are all of one sign, and the coefficients of the odd powers are all of opposite sign, then the equation has no negative root. Hence, the equation 6x4 - 11x3 + 5x2 - 2x + 1 = 0 has no negative root
  • If the equation contains only even powers of x and the coefficients are all of the same sign, the equation has no real root. Hence, the equation 4x4 + 5x2 + 2 = 0 has no real root.
  • If the equation contains only odd powers of x, and the coefficients are all of the same sign, the equation has no real root except x = 0. Hence, the equation 5x5 + 4x3 + x = 0 has only one real root at x = 0.
  • Descartes' Rule of Signs : An equation f(x) = 0 cannot have more positive roots than there are changes of sign in f(x), and cannot have more negative roots than there changes of sign in f( - x). Thus the equation x4 + 7x3 - 4x2 - x - 7 = 0 has one positive root because there is only change in sign. f( - x) = x4 - 7x3 - 4x2 + x - 7 = 0 hence the number of negative real roots will be either 1 or 3.

   roots

EXAMPLES:

theory of equations problems

I shall have to end here and leave the rest of it for my CBT Club students. I shall cover some problems based on this in the CBT Club this week.

 

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Re: Theory of Equations..
by shruti a - Thursday, 28 June 2007, 10:23 AM
 

hi

  • In eg 5 above, shouldnt the answer be all of these instead of none of these?
  • Descartes’ Rule of Signs : An equation f(x) = 0 cannot have more positive roots than there are changes of sign in f(x), and cannot have more negative roots than there changes of sign in f( - x). Thus the equation x4 + 7x3 − 4x2 − x – 7 = 0 has one positive root because there is only change in sign. f( - x) = x4 − 7x3 − 4x2 + x – 7 = 0 hence the number of negative real roots will be either 1 or 3.

      why cant the number of negative roots be 2 also?

thanks

 

Re: Theory of Equations..
by Rohit Suresh - Friday, 29 June 2007, 04:30 PM
 

@ shruti,

             Since the degree of the equation is 4, it has to have 4 roots either real or imaginary. Since only one root can be positive at the max, the other three can either be

i) all negative and real 

OR

ii) one negative and real and the other two imaginary, since imaginary roots always exist in pairs.

So there cant be 2 negative 'real' roots alone. The word real is very important here. Hope you got it.

Re: Theory of Equations..
by Amishe 800 - Tuesday, 10 July 2007, 04:56 AM
 

TG ,

Your article says ...the sum of roots for the eq .

   nature of roots

The sum of roots should be =  -1 * a1/a0..

Please confirm

 

Re: Theory of Equations..
by Sodium Hydro Phosphate - Friday, 10 August 2007, 12:23 PM
 

What are you asking Amishe 800?

Isn't -1*a1/a0 = -a1/a0 ?

Its 4.56 am that you have written this. I guess u needed some sleep now smile yeah?

Re: Theory of Equations..
by lavika gupta - Saturday, 11 August 2007, 08:08 AM
  hi tg.please include some more algebra topics like maxima & minima,etc.
Re: Theory of Equations..
by richa chopra - Monday, 10 September 2007, 09:43 PM
 

hey tg

in example 2 shouldnt d answer be 5 .....

btw d article is amazing...

 

Re: Theory of Equations..
by Ranvijay Singh - Thursday, 13 September 2007, 05:26 PM
  Hi Richa,

I think both the answers of example #2 - 2 as the sum of the squares of the roots and 11 as the sum of the cubes of the roots are correct. TG has already explained it. You probably missed something while calculating/substituting the values. It's fairly simple and without skipping few redundant steps, the solution may look like this:

Given equation: x3 - 2x2 + x - 3 = 0. Let a, b, c are the roots of this equation.
So, a + b + c = - (-2/1) = 2  (using the formula -a1/a0)
ab + bc + ca = 1/1 = 1        (using the formula a2/a0)

Now, (a + b + c)2 = a2 + b2 + c2 + 2 (ab + bc+ ca)
=> a2 + b2 + c2 = (a + b + c)2 - 2 (ab + bc + ca)
=> a2 + b2 + c2 = 4 - 2 = 2 (sum of the squares of the roots)

Now, substituting the roots in the original equation, we get three equations:
a3 - 2a2 + a - 3 = 0 ... (i)
b3 - 2b2 + b - 3 = 0 ... (ii)
c3 - 2b2 + c - 3 = 0 ... (iii)

Adding these three equations, we get
(a3 + b3 + c3) - 2(a2 + b2 + c2) + (a + b + c) - (3 + 3 + 3) = 0
=> (a3 + b3 + c3) - 2 * 2 + 2 - 9 = 0
=> a3 + b3 + c3 - 11 = 0
=> a3 + b3 + c3 = 11  (sum of the cubes of the roots)


~Vijay


Re: Theory of Equations..
by Richa Mittal - Thursday, 13 September 2007, 05:39 PM
 

hello sir,

i would like to know that in ques no 6, how can u cancel ( x-1) on both sides,since it is makin the euation zero(0).

 

Re: Theory of Equations..
by Ranvijay Singh - Thursday, 13 September 2007, 08:58 PM
  Hi Richa,

In question 6, we first cancel (x - 1) from both the sides and then put x = 1. Of course, for x = 1, the division isn't defined. After getting the term canceled out from both the sides, the equation reduces to (x - a)(x - b)(x -c)... = (x9 + x8 + ... + 1). Now, we can easily put x = 1 to get the required value as it's a equation in x and for x = 1, we can always find some a, b, c, ... , such that (1 - a)(1 - b)(1 - c)... = 10.

The confusion is probably because we are using 1 in both the steps - canceling (x - 1) and putting x = 1. But, these two are independent of each other. Had it been (2 - a)(2 - b)..., we would have simply put x = 2 to get the required value.


~Vijay
Re: Theory of Equations..
by garima jain - Tuesday, 18 September 2007, 07:48 PM
  please give in more chapters on algebra including graphs..smile
Re: Theory of Equations..
by garima jain - Tuesday, 18 September 2007, 08:15 PM
  please give in more chapters on algebra including graphs..smile
Theory of Equations..
by himali agarwal - Friday, 19 October 2007, 05:53 PM
 

Hi TG

when it is written that maximum no of negative roots is equal to the change in sign of f(-x), then in the e.g. given, why are we considering 1 or 3 number of negative roots. It should be 1 or 2. Even if we say that it would be having a pair of complex roots, the we should write 1 only as the number of negative roots as 2 or 3 roots cannot be possible. Please explain?

Re: Theory of Equations..
by Guruprasad SP - Wednesday, 28 May 2008, 08:28 PM
 

Sir/Friends plz help

If the roots of the equation , ax^2+bx+c=0, are of the form

p/p-1 and p+1/p then the value of (a+b+c)^2 is

Options: b^2-2ac ,  b^2-4ac ,  2b^2- ac ,   4b^2-2ac

ans : b^2-4ac

Re: Theory of Equations..
by raja thumma - Wednesday, 4 June 2008, 08:06 PM
 

sum of roots -b/a =2p^2-1/p(p-1)

product of roots c/a =p+1/p-1

(a+b+c)^2=a^2(1/p(p-1))^2

diff of roots for quadratic eq  b^2 -4ac =a^2(1/p(p-1))^2

x= -b +(b^2 -4ac )^0.5 /2a

Re: Theory of Equations..
by Dhruv Rakesh - Thursday, 5 June 2008, 12:17 AM
  one cannot just "cancel" x-1, when x is being put as 1 later in the question, as that is an undefined operation.... a better way to look at it is....
(x10-1)/(x-1) is nothing but the sum of the geometric series (1+x+x2+.....till x9)

so we don't cancel. Rather we change its form.


Re: Theory of Equations..
by Ankurkumar Bhatt - Monday, 9 June 2008, 06:30 PM
 

Hi Vijay !!

 

Thank you for the clarification, at first I didn't understand the main solution, you made it easy when you added all the roots to get cubes summatino. Thanks again....

 

Ankur Bhatt

Re: Theory of Equations..
by Total Donkey - Wednesday, 25 June 2008, 11:13 PM
  hello TG,

The point raised about substituting the value of x=1 is a valid one ....

e.g  if we have an equation F(x) . (x-1) = G(x) . (x-1)

(just to remind my friends, we actually dividing both the equations with (x-1) to get the simplified version which isn't valid for x=1 because division by Zero is not allowed)

we can conclude that F(x) = G(x) when x != 1

so we can't say that F(1) = G(1) ...

To make this point more lucid, consider the functions:

F(x) = 1 when x!=1
      = 0 when x=1

G(x) = 1 for all values of x

Now the equation F(x) . (x-1) = G(x) . (x-1) holds for all values of x
but surely we can't get the value of F(1) from G(1)

waiting for your reply

Re: Theory of Equations..
by Mech Guy - Wednesday, 2 July 2008, 10:00 PM
 

can anybdy help me out in questions such as

for how many integral values of x does the eq. (ax2 + bx +c)/ (dx2+ex+f) has integral values...??

There is a question in the number system problems...!!

 

Thanks

mechguy

Re: Theory of Equations..
by manish sharma - Thursday, 10 July 2008, 01:01 PM
  for how many values of a, a>0, both the roots of ax^2-(a+1)x+(a-2)=0 are greater than 3?


for both roots to be greater than 3, sum sholud be > 6.  check this condition and no solution will be there.

now one might thing that product sholud also be>9. but in this case roots can be negative also. as -9 * -8 = 72>9. both roots are lesser than 3 here..

nice one..!!!!
Re: Theory of Equations..
by nitin . - Tuesday, 5 August 2008, 11:16 PM
  Hi TG sir,

excellent article....

Hoping same for graph theory..Please come up with it.

Thanks a lot!

Re: Theory of Equations..
by Jegathiesh P - Wednesday, 6 August 2008, 01:42 PM
 

Hi TG and others,

  For example No:4, isn't the answer is none of these. Since the equation,   f(x) must have to be equalled to zero at x=any of the roots.

i.e  f(x)=0 at x=a,b,c if the f(x) is of degree 3, where a,b,c are the roots of the equation f(x).

 Please correct me if I am wrong.

Thanks

 Jegathiesh.P

Re: Theory of Equations..
by Jegathiesh P - Wednesday, 6 August 2008, 02:00 PM
 

Hi Guru,

Assume some values for p, take p=5. So the roots are 5/4 and 6/5. With the roots, frame the equation. (x-5/4)(x-6/5), you will get the equation as,   20x 2 - 49x+30=0 where a=20, b=-49 and c=30. So (a+b+c)^2 would be     (20-49+30)^2=1. Also, try b^2-4ac=(-49)^2-(4x30x20)=2401-2400=1. Hence, the answer should be b^2-4ac.

Let me know if any concerns.

Regards,

Jegathiesh.P

Re: Theory of Equations..
by nice Smile - Sunday, 14 June 2009, 06:46 PM
  hi tg,

i m posting this question here since i could not understand how u expanded  (1 - x)-4 in the equation (1 - x9)(1 - x30 - 3x10 + 3x20)(1 - x)-4 to get the co-efficient of x11. this seems to be related to this topic, hence posting here
Re: Theory of Equations..
by yogesh bansal - Monday, 22 June 2009, 12:14 AM
 

hi nice smile..

i hv jst joind tg..Can u write original post...i thnk i cn help u..ds is my interest area..

Re: Theory of Equations..
by nice Smile - Sunday, 19 July 2009, 08:03 PM
  to get the place where this discussion was done please refer the last few posts in the Groupings and distributions chapter.

I have one doubt here.

i think the answer option for question number 4 should be "all of these", since none among a, b and c are the roots of the equation
2/3,1/6,1/3 are positive so all of these are not the roots of the equation. so the answer should be all of these since the question asks which of the following are not the roots of the equation.

I know that this was the first question in this thread, but still asking this since there was no reply for this.

Please correct me if my understanding is wrong.

Re: Theory of Equations..
by gaurav kaushal - Sunday, 2 August 2009, 11:02 AM
  what is  –    ???????
Re: Theory of Equations..
by - SK - Sunday, 2 August 2009, 07:57 PM
 

 

Fantastic article.. Truly helpful and handy one sir ..
thanks a lot

-Shiva

gaurav > negative sign in the overall article is being shown as  â€“  


 

Re: Theory of Equations..
by ankur rana - Tuesday, 4 August 2009, 05:24 PM
  hello everyone
how many sign changes are there in
x3+x2-4x+4=0{x3=x raised to the power 3}
one or two
one from +ve to -ve and 2nd from -ve to +ve?
Re: Theory of Equations..
by jaya arora - Wednesday, 5 August 2009, 11:52 AM
 

answer 1

as no. of sign changes on f(-x) is 1

Re: Theory of Equations..
by Prashant Sharma - Thursday, 13 August 2009, 12:34 AM
 

Hello friends,

Can anyone help me with correct approach for solving following:

x^4-256x^3+kx^2-496x-2008 = 0 has product of two of the roots =8.Find the value of k.

 

Thanks in advance!!

Re: Theory of Equations..
by baby assassin - Wednesday, 2 September 2009, 06:58 PM
  is the question right ...494 instead of 496 will solve the problem
Re: Theory of Equations..
by Netra Mehta - Friday, 4 September 2009, 12:18 PM
  Hi TG!
Plz help me in solvin d followin question...

The maximum possible value of x^2 + 4y^2 + 9z^2, subject to x+2y+3z=12, where x,y,z are real numbers, is
A. 48 B.224 C.240 D.140 E.Infinite
Re: Theory of Equations..
by saurabh aggarwal - Friday, 4 September 2009, 02:31 PM
  thanx.....it was very helpful
Re: Theory of Equations..
by Chowparan Saha - Friday, 4 September 2009, 02:54 PM
  Hi TG Sir !!,
Another great fan of yours. hats off to you for your tremendous effort. It would really help to hear some of your enlightenments on PROBABILITY big grin
Re: Theory of Equations..
by Raju Singh - Monday, 28 September 2009, 04:15 PM
  Thanks TG for your great efforts.
Descartes’ Rule of Signs :
Can any one tell me what this equation is all about
x4 + 7x3 − 4x2 − x – 7 = 0

Plz tell me abt these special characters used in above eq.
In simpler character plz write that equ.
I got to knw about meaning of – as -ve sign. What abt others??

Re: Theory of Equations..
by Pallav Jain - Sunday, 4 October 2009, 08:27 PM
  Can any one tell me the formula for
a^3 +b^3 +c^3 =
in eg. 2

Re: Theory of Equations..
by Arjun Nair - Tuesday, 6 October 2009, 12:55 PM
 

he hasnt used a formula ...

he substituted the roots(alpha,beta,gamma) in the equation itself and then added the 3 resulting equations ...

then replaced the resulting grouped terms with there values

Re: Theory of Equations..
by aashish arora - Saturday, 10 October 2009, 09:24 PM
  hi netra


i guess u it shud be mentioned here that x,y,z are positive real numbers
otherwise its a futile question


if my guess is right,then the answer is 48


if not......i wud guess "infinite"



regards
aashish
Re: Theory of Equations..
by jai singh - Sunday, 11 October 2009, 11:14 AM
  awwwweeeeeeesome
Re: Theory of Equations..
by nice Smile - Tuesday, 20 October 2009, 10:57 PM
  can somebody help me

Please explain how to expand

(1 - x9)(1 - x30 - 3x10 + 3x20)(1 - x)-4 and get the co-efficient of x11 in this.
Re: Theory of Equations..
by Total Gadha - Wednesday, 21 October 2009, 09:44 PM
  expansion of (1 + x)n = 1 + nx + n(n - 1)x2/2 + n(n - 1)(n - 2)x3/6 + ...

expand (1 - x)-4 this way...
Re: Theory of Equations..
by barry white - Saturday, 18 September 2010, 10:39 AM
  Great article... Such a lucid explantion... Ur awesome..

cheers
Re: Theory of Equations..
by ANSHUMAN SANGURI - Monday, 20 September 2010, 01:54 AM
  i dont knw if the moderators stil go over these articles to check fr nw comments bt seriusly these tg articles r really cut above the rest .
i m closely following thm fr some tim nw as i m arming myslf with cat solvin armoury,these articls on quant and verbal are jst wht one is lukin fr.........
thnks ......
Re: Theory of Equations..
by Vinu Varghese - Tuesday, 21 September 2010, 01:46 AM
  Beautiful.... lucid ... really short of words..... cleared a lot of my doubts..... keep te good wwork
Re: Theory of Equations..
by Ayshwar Pandey - Tuesday, 19 October 2010, 06:30 AM
  In the last question, u cancelled (x-1) from both sides which in fact is 0. Isn't this wrong?
Re: Theory of Equations..
by mohil mittal - Thursday, 15 September 2011, 09:06 AM
  In last question if we put 1-x instead of x then the roots of the equation would be 0,1-alpha, 1-beta,......
Calculate the product of roots with 9 items that is 10C9=10


Re: Theory of Equations..
by sonnel singh - Wednesday, 21 September 2011, 03:22 AM
  can someone plz answer these questions:-

Q1) If the roots of equation x^3 - a.x^2 + b.x - c = 0 are three consecutive integers, then what is the smallest possible value of b??

Q2) If the equation x^3 - a.x^2 + b.x - a = 0 has 3 real roots, then it must be the case that

1) b = 1
2) b not= 1
3) a = 1
4) a not= 1
Re: Theory of Equations..
by TG Team - Thursday, 22 September 2011, 05:19 PM
 

Hi Sonnel smile

1. Let's say three consecutive integral roots are ß - 1, ß and ß + 1, then b = (ß - 1)ß + ß(ß + 1) + (ß - 1)(ß + 1) = 3ß² - 1 where ß is an integer.

So minimum value of b = -1 when ß = 0.

2. If b = 1, then equation becomes x³ - ax² + x - a = (x² + 1)(x - a) = 0 i.e. all the three roots can't be real.

That ensures that b should not be equal to 1.

Kamal Lohia 

 

Re: Theory of Equations..
by sonnel singh - Sunday, 9 October 2011, 02:19 AM
  thnx alot big grin smile
Re: Theory of Equations..
by Shashank Pandey - Tuesday, 11 October 2011, 07:25 PM
  In example 4,option d should be all of these, as the given equation 6x^3 + 5x^2 + 2x + 2=0 can never have a positive root. Am i right moderators? smile
Re: Theory of Equations..
by aakansha rawat - Saturday, 5 November 2011, 06:32 PM
 

do we have a method to find the general equation of a chord of a given equation of a curve????????????????????????????????????//

Re: Theory of Equations..
by suraj saxena - Monday, 8 October 2012, 04:51 PM
  hi,

Sir how to solve equations of the type

eg : find no of positive integral solutions of 7x+12y+4z=30

SOORAJ
Re: Theory of Equations..
by subramaniam gounder - Friday, 18 January 2013, 11:18 PM
  in question 4 since all the coefficient are positive then all the roots are not positive. so the options have oly positive roots... hence the 4th option should be all of these... right..???
Re: Theory of Equations..
by yash modi - Monday, 30 May 2016, 10:31 AM
  Hey sir,
I Want to know to solve this question


\"If f(x) = x^2+2x-5 and g(x) = 5x+30, then the roots of quadratic equation g[f(x) will be?\" ... Here what does g[f[x] mean?
Re: Theory of Equations..
by yash modi - Monday, 30 May 2016, 02:42 PM
  Hey sir,
I Want to know to solve this question


\"If f(x) = x^2+2x-5 and g(x) = 5x+30, then the roots of quadratic equation g[f(x) will be?\" ... Here what does g[f[x] mean?