How do determine the factors of an equation having degree higher than 2?

Question

e.g. what are the factors of eqauation x^3-6x^2+11x-6=0
Expalin the procedure.

Answer 1

There are calculators that do this sort of thing, of course.

Algebra textbooks cheat. They give students a great many problems that factor neatly. In real life, those factors are much rarer, and it is necessary to learn other techniques.

Here are a couple.

Check first to see if x = 1 is a root. It often is, at least in high school, and checking is as easy as seeing if the coefficients of the equation add up to zero. That works for the equation you posted. It isn't much harder to see if -1 is a root. [It isn't.]

Suppose that didn't work...

A cubic has three roots. It is quite possible that one of the roots is considerably smaller in magnitude than the other two. In that case, the biggest root, cubed, will be large enough that the constant term in the equation can be ignored. Let's try this with

2x³ -15x² - 53x + 30 = 0

"Ignoring" the 30 leaves

x ( 2x² - 15x - 53 ) = 0

Using the quadratic formula with 2x² - 15x - 53 = 0 yields
x = 10.1. Aha! Let's try x = 10. And it works. Even if the true root is a little different from being exactly 10, it is much easier to find, graphically, say, than if you have no idea where to look.

The reverse process may work as well. Suppose a root of the equation is less than 1. Then x³ will be much smaller than the other terms and might be ignored. Let's try that. If x³ is ignored, the equation becomes

-15x² - 53x + 30 = 0

This time the quadratic formula yields x = 0.496. Let's try x = ½. Yep!

Now that we know that x = ½ and x = 10 are roots, we can write

2x³ -15x² - 53x + 30 = (2x - 1)(x - 10)(x - r)

Since (-1) times (-10) times (-r) must equal 30, r = -3, and that is the third root.

Can you write a simple computer program? If not, stop here, but if you can, try this:

-- Define the function you wish to solve: f(x) = 0

on f(x)
    set fun to x ^ 3 - 6 * (x ^ 2) + 11 * x - 6
    return fun
end f

-- Estimate the slope of f(x):

on fp(x)
    set fplus to (f(x + 0.01 * x) - f(x - 0.01 * x)) / (2 * 0.01 * x)
    return fplus
end fp

-- Guess a value for x

set x to 5
repeat with i from 1 to 10
    set xnew to x - f(x) / (fp(x))
    set errorT to xnew - x
    if errorT ^ 2 < 1.0E-10 then
        display dialog ("x = " & xnew)
        exit repeat
    end if
    set x to xnew
end repeat

An initial guess of x = 5 leads to the root x =3. Now you can divide ( x³ - 6x² + 11x - 6 ) by (x -3) to get x² - 3x +2, which factors into (x -1)(x-2) with roots of x =1 and x = 2.

Hope this tutorial is of help. The numerical method used here is called "Newton-Raphson" and it is described many places on the Internet. Just go to Google and look for Newton-Raphson.

Answer 2

Using the Remainder Theorem:
Let f(x) = x^3-6x^2+11x-6.
f(1) = 1-6+11-6 = 0
So (x-1) is a factor.

Let f(x) = x^3-6x^2+11x-6 = (x-1)(Ax^2 + Bx + C)
Using Comparing Coefficients:
x^3: 1 = A
constant: -6 = -C => C = 6
x^2: -6 = B - A => -6 = B - 1 => B = -6+1 = -5

So f(x)
= x^3-6x^2+11x-6
= (x-1)(x^2 - 5x + 6)
= (x-1)(x-2)(x-3)

Answer 3

Let f(x) = x^3-6x^2+11x-6.

f(1) = 1-6+11-6 = 0
So (x-1) is a factor.

Let f(x) = x^3-6x^2+11x-6 = (x-1)(Ax^2 + Bx + C)

Comparing Coefficients:
Coef. of x^3: 1 = A
constant: -6 = -C => C = 6
Ceff. of x^2: -6 = B - A => -6 = B - 1 => B = -6+1 = -5

So f(x)
= x^3-6x^2+11x-6
= (x-1)(x^2 - 5x + 6)
= (x-1)(x-2)(x-3)

Answer 4

Use synthetic division to test for possible rational roots. Rational roots will be factors of the constant/factors of the coefficient of the highest ordered term. In your example the factors of -6 are ±6,±3,±2,±1 and for the highest coefficient ±1 so possible rational roots are the same as the factors of the 6.
If none of those works, then check for sign changes in the remainder of each division. Real roots will be between the factors where a sign change occurred. By successive approximation you can zero in on the real root to get an approximate answer to whatever precision you want.

Answer 5

put F(1) abov eqn will be zero then x-1 is a factor
divd x^3-6x^2+11x-6=0 by x-1 then u will get another factors

Answer 6

synthetic division
double synthetic division
trail and error
roots
remainder theroem