Thread: working backwards - cubics

Write an equation that has the following roots: 2, -1, 5

Answer key: x^3 - 6x^2 + 3x + 10 = 0

For quadratic equations, I use the sum and product of roots, this is a cubic equation, how do I solve this?

Thanks.

Originally Posted by shenton

Write an equation that has the following roots: 2, -1, 5

Answer key: x^3 - 6x^2 + 3x + 10 = 0

For quadratic equations, I use the sum and product of roots, this is a cubic equation, how do I solve this?

Thanks.

Thanks! That turns out to be not as difficult as imagined. I thought I needed to use sum and products of roots to write the equation, it does makes me wonder a bit why or when I need to use sum and products of roots.

Write an equation that has the following roots: 2, -1, 5

Is there any other way to solve this other than the (x-2)(x+1)(x-5) method?

If we have these roots: 1, 1 + √2, 1 - √2

the (x - 1) (x -1 -√2) (x -1 +√2) method seems a bit lenghty.

When we expand (x - 1) (x -1 -√2) (x -1 +√2) the first 2 factors,

it becomes:

(x^2 -x -x√2 -x +1 +√2) (x -1 +√2)

collect like terms:

(x^2 -2x -x√2 +1 +√2) (x -1 +√2)

To further expand this will be lenghty, my gut feel is that mathematicians do not want to do this - it is time consuming and prone to error. There must be a way to write an equation other than the above method.

Is there a method to write an equation with 3 given roots (other than the above method)?

Thanks.

Originally Posted by shenton

Write an equation that has the following roots: 2, -1, 5

Is there any other way to solve this other than the (x-2)(x+1)(x-5) method?

If we have these roots: 1, 1 + √2, 1 - √2

the (x - 1) (x -1 -√2) (x -1 +√2) method seems a bit lenghty.

When we expand (x - 1) (x -1 -√2) (x -1 +√2) the first 2 factors,

it becomes:

(x^2 -x -x√2 -x +1 +√2) (x -1 +√2)

collect like terms:

(x^2 -2x -x√2 +1 +√2) (x -1 +√2)

To further expand this will be lenghty, my gut feel is that mathematicians do not want to do this - it is time consuming and prone to error. There must be a way to write an equation other than the above method.

Is there a method to write an equation with 3 given roots (other than the above method)?

Thanks.

You have a pair of roots of the form a+sqrt(b) and a-sqrt(b) if you multiply
the factors corresponding to these first you get:

(x-a-sqrt(b))(x-a+sqrt(b))=x^2+(-a-sqrt(b))x+(-a+sqrt(b))x +(-a-sqrt(b))(-a+sqrt(b))

................=x^2 - 2a x + (a^2-b)

Which leaves you with the easier final step of computing:

(x-1)(x^2 - 2a x + (a^2-b))

RonL

Hello, shenton!

The sum and product of roots works well for quadratic equations.

For higher-degree equations, there is a generalization we can use.

To make it simple (for me), I'll explain a fourth-degree equation.


Divide through by the leading coefficient: .

Insert alternating signs: .
. . . . . . . . . . . . . . . . . .


Suppose the four roots are:

The sum of the roots (taken one at a time) is:
. .

The sum of the roots (taken two at a time) is:
. .

The sum of the roots (taken three at a time) is:
. .

The sum of the roots ("taken four at a time") is:
. .

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

For your problem with roots:
. . we have: .

Then: .
. . Hence:

And: .
. . Hence:

And: .
. . Hence:


Therefore, the cubic is: .

This is awesome, Soroban. Using the method you shown, I was able to solve this problem:

1, 1+√2, 1-√2

a=1, b=1+√2, c=1-√2

Let x^3 - px^2 + qx - r = 0 be the cubic equation

p = a + b + c
= (1) + (1 + √2) + (1 - √2)
= 3

q = ab + bc + ac
= (1)(1 + √2) + (1 + √2)(1 - √2) + (1)(1 - √2)
= 1 + √2 + 1 - 2 + 1 - √2
= 1

r = abc
= (1)(1 + √2)(1 - √2)
= 1-2
= -1

Therefore x^3 - px^2 + qx - r = 0 becomes
x^3 - 3x^2 + x - (-1) = 0
x^3 - 3x^2 + x + 1 = 0

Thanks for the help and detailed workings.