Friday
April 18, 2014

Homework Help: Factoring

Posted by Ace on Wednesday, May 30, 2007 at 12:47am.

What does 3x^3-3x+2=0 come out to? How would I solve this?

Is the x cubed supposed to be x squared?

You can compare the equation with the identity:

(a+b)^3 = a^3 + 3 a^2 b + 3 a b^2 + b^3

which you can rewrite as:

(a + b)^3 = 3 ab (a + b) + a^3 + b^3 (1)

The equation you want to solve is:

x^3 = x - 2 (2)


Equation (1) is an identity that is always true. I.e., whatever you substitute for a and b, the equation is always satisfied. Equation (2) has, of course, at most three solutions. Suppose you choose a and b such that:

3 ab = 1 (3)

and

a^3 + b^3 = -2 (4)

then equation (1) says that:


(a + b)^3 = (a + b) - 2

and you see that a + b would then satisfy equation (2). So, how do we solve equations (3) and (4) for a and b? You take the cube of equation (3):

3 ab = 1 --->

a^3 b^3 = 1/27

If you put A = a^3 and B = b^3, then:

A B = 1/27 (5)

equation (4) can be written as:

A + B = -2 (6)

If you use (6) to express B in terms of A and substitute that in (5) yo obtain a quadratic equation for A. Solve that equation. You obtain two solutions. Because of the symmetry between A and B (i.e. if you interchange A and B in (5) and (6) the two equatons remain the same), you can take A to be one of the solutions and B to be the other solution.

Extract the cube roots to obtain a and b, add them up to obtain the solution. To obtain the two other solutions, you must use that for given A there are three solutions for a, such that

a^3 = A:

a = A^(1/3) Exp[2 n pi i/3]

where n is 0, 1 or 2. The same is true for b, but we need to satisfy
equation (3). This means that if we choose a to be:

a = A^(1/3) Exp[2 n pi i/3]

then b is given by:

b = B^(1/3) Exp[-2 n pi i/3]

Answer this Question

First Name:
School Subject:
Answer:

Related Questions

algebra - x cubed + x squared over x squared - 16 times x + 4 over 3x to the ...
maths - If A=1+r^a+r^2a+r^3a+.......................to infinity B=1+r^b+r^2b...
math - just an algebra question. how would you solve: (3x^2 +5)(3x^2 +5) ^ means...
calculus - If (x) =3x squared - x cubed, find f prime(1) and use it to find an ...
math,help - Is this trinomial a perfect square x^2-6x+9 if the trinomial starts ...
Pre-calc, Algebra, Trig - (2x squared-3x +1)(4)(3x+2)cubed (3)+(3x +2)raised to ...
calculus - I am trying to find equation of the tangent line to the curve 2/(1-3x...
math 123 - solve the equation by factoring 3x^2-8x-3=0 (3x+1)(1x-3)=0 Now solve ...
algebra - Solve the system by graphing. 3x+y=6 3x-y=0 I'm kind of unsure about ...
algebra 2 - I am so stuck on this factoring thing. how do you solve: 1)15a(...

Search
Members