Posted by **mary** on Sunday, January 21, 2007 at 6:23pm.

Still looking at how to solve these...

Both are cubic polynomials?

43x^3 + 13x^2 + 18x - 456 = 0

DO I divide first by x-1?

2x(3x+2)^2=312

Honestly: I would graph the equations and find the roots first. I don't see them off hand.

graph y=43x^3 + 13x^2 + 18x - 456

and

y=2x(3x+2)^2 -312

Try to find the rational roots (if there are any).

If the equation

43x^3 + 13x^2 + 18x - 456 = 0

has rational roots x = p/q then p must be a divisor of 456 and q must be a divisor of 43. I didn't find anything , but I only tried for half a minute :)

If you find a rational root, say x = y, then you divide the polynomial by x-y and solve the resulting quadratic equation.

If there really aren't any rational roots you can solve the cubic equation exactly as follows.

First you substitute:

x = y - 13/3

and work out the polynomial as a function of y. This has the effect of eliminating the quadratic term. So, you get an equation of the form:

y^3 + p y + q = 0 (1)

You solve this equation by comparing this equation to the equation for

(A + B)^3:

(A + B)^3 = A^3 + 3 A^2B + 3 A B^2 + B^3

rearranging gives:

(A + B)^3 - 3AB(A + B)-(A^3 + B^3) = 0

What use is this? Well, suppose you can find A and B such that

3AB = -p (2)

and

A^3 + B^3 = -q (3)

then Y = A + B would satisfy the equation (1)

This should be possible because we need to solve two equations for two unknowns A and B. So, how do we go about solving for A and B? What you do is you take the third power of Eq. (2):

27 A^3 B^3 = -p^3

If you now put A^3 = s and B^3 = t then this means that

s*t = -p^3/27

while Eq. (3) implies that:

s + t = -q

Combining these two equations for

s and t gives you a quadratic equation for these quantities. By extracting the cube root you find A and B,. Add them together and you find y. Subtract the term 13/3 to find x.

It turns out that you need to make use of complex numbers to find all three roots even if they are all real.

## Answer This Question

## Related Questions

- MAth - Find X 43x^3+13x^2+18x = 456 2x(3x+2)^2 = 312
- math - Still having problems solving this... 2x(2x + 1)^2 = 312 I started with: ...
- Math - Solve the following equation. 4.7 = 1.13x^2 Do i square root the 1.13x^2 ...
- Algebra - To solve the following system of equations by elimination, which step...
- polynomials naman - If remainder on division x^3-kx^2+13x-21 by 2x-1 is -21 ,...
- Math - Roots Ok, what about roots? Roots of polynomials? Square roots? Cube ...
- math,correction&help - Is this right:if it still needs more can someone help me ...
- math - please anwer these: 1) how can i multiply and divide polynomials ...
- Cubic Equations - I have three that I need help with if possible. 1. Solve 2x^3...
- Math-find roots - 1.One root of each equation below is the same for both ...

More Related Questions