- #1
lionely
- 576
- 2
Homework Statement
Find the equation who roots are the squares of the roots of the equation x^3-4x^2+x-
is the answer
x^3-16y^2+y+1?
Neither of these is an equation.lionely said:Homework Statement
Find the equation who roots are the squares of the roots of the equation x^3-4x^2+x-
is the answer
x^3-16y^2+y+1?
lionely said:I know i took the roots of the 1st equation to be x so the ones for the new equation are x^2.
I said let y= new roots, y= x^2
x= sqrt y
and I put x in first equation and I got x^3-16x^2+x+1=0
lionely said:x^3-4x^2+x-1=0
new roots = x ^2
let y=x^2, x = sqrt y
(sqrty)^3-4(sqrty)^2+sqrty - 1
(y^3/2 - 4y + sqrty -1)=0
squaring both sides
y^3+16y^2+y+1
lionely said:D:, I'll leave it as (y^3/2 - 4y + sqrty -1)=0 then.
lionely said:Is it a^6+3 a^4 y+7 a^4+3 a^2 y^2+14 a^2 y-5 a^2+y^3+7 y^2-5 y-1?
lionely said:Isn't it
y= x-a^2
so x = y+a^2? then i put it into the equation?
Dick said:I think you missed the lecture were they were actually talking about how to do this problem. Sure, you could do that, but it's not a polynomial and it's not particularly well defined either if all of the roots aren't positive. What you missed is that if your polynomial has three roots a,b,c then you can write it as (x-a)(x-b)(x-c) then the coefficients of your original polynomial are various symmetric polynomials of a,b and c. The polynomial you want is (x-a^2)(x-b^2)(x-c^2). The coefficients are also symmetric polynomials of a, b, and c. Doesn't that ring some kind of bell?
lionely said:Is it √x(x+1) = 1 + 4x ?
lionely said:I know i took the roots of the 1st equation to be x so the ones for the new equation are x^2.
I said let y= new roots, y= x^2
x= sqrt y
and I put x in first equation and I got x^3-16x^2+x+1=0
lionely said:x^3-4x^2+x-1=0
new roots = x ^2
let y=x^2, x = sqrt y
(sqrty)^3-4(sqrty)^2+sqrty - 1
(y^3/2 - 4y + sqrty -1)=0
squaring both sides
y^3+16y^2+y+1
lionely said:so is it ... {√x(x+1)}2 = 1 + 16x2?
Curious3141 said:Your method is basically sound, but I'm betting you screwed up the algebra. I did it your way and got the right answer. You have to be very careful in grouping terms and squaring out the surd forms.
While others have suggested using a slightly more involved method to compare the sums/products of roots to the coefficients, this is unnecessary here.
Dick said:You are right. There is an easier approach. Nice.
lionely said:SIGH , okay (y3/2 - 4y + √y -1)
= y.√y -4y + √y -1
= y.√y + √y = 1+ 4y
= √y(y+1) = 1 + 4y
then i square both sides?
so i get
y(y+1)2 = 1 + 16y^2
Curious3141 said:Thanks, yes, this is an elementary approach, but the OP is messing up the algebra and failing to see why. It's the simple binomial square that he's getting wrong.
lionely said:sighh sigh sighj sigh sigh sigh sigh why didn't i see that ...
(1+4x)^2 = 1 + 8x + 16x^2...
lionely said:it is x3 -14x2+x-1
lionely said:lol... x^3 -14x^2-7x-1
The theory of cubic equations is a mathematical concept that deals with finding the roots of a cubic polynomial equation. It involves understanding the properties and characteristics of cubic equations and using various methods to solve them.
There are several methods for solving cubic equations, including the rational root theorem, synthetic division, and the cubic formula. The most commonly used method is the cubic formula, which involves plugging in the coefficients of the equation into a formula to find the roots.
A cubic equation is a polynomial equation of degree 3, while a quadratic equation is a polynomial equation of degree 2. This means that a cubic equation has three solutions or roots, while a quadratic equation has two. Additionally, the methods used to solve these equations are different.
Yes, all cubic equations have at least one real root. However, some cubic equations may have complex or imaginary roots, which may not be relevant in certain real-world applications.
Cubic equations have many real-life applications, including in physics, engineering, and economics. They can be used to model the motion of objects, calculate the volume of a cube, and predict the growth of a population over time.