problem with symbolic factorization with two symbols

The output is not a factorized one. I want this form : (s+s1)*(s+s2)*(s+s3). What is the problem?

syms s K

factor(s^3 + 10*s^2 + (21+K)*s + 4*K, [s, K],'FactorMode','full')

The output is

4*K + 21*s + K*s + 10*s^2 + s^3

[Dyuman Joshi]

It is factorized according to the inputs given.

From the documentation - F = factor(x,vars) returns an array of factors F, where vars specifies the variables of interest. All factors not containing a variable in vars are separated into the first entry F(1). The other entries are irreducible factors of x that contain one or more variables from vars.

Note the last sentence.

What is the expected output from the vectorization?

Answer 1

[John D'Errico]

That you want to see a simple set of factors is not relevant. You essentially need to obtain the roots of this cubic polynomial in s, where K is a parameter, which would give you that factorization.

syms s K
sroots = solve(s^3 + 10*s^2 + (21+K)*s + 4*K,s,'maxdegree',3)

sroots = 

As you can see, they are fairly messy, and depending on the value of K, they may well be complex even if K is real. This is expected for any cubic polynomial, that we may find exactly two conjugate complex roots.

The factors now are just

prod(s-sroots)

ans = 

We can expand that to show the result recovers your original polynomial.

simplify(expand(ans))

ans = 

Finally, why did not factor understand what you wanted to see? You gave it two variables s and K. How should factor possibly know that you wanted it to do what I just did, essentially, that s is the variable of interest, and K a symbolic parameter in the problem? Software cannot read your mind, at least, not yet.

[Torsten]

It also doesn't seem to work if you only specify s instead of [s,K] in

syms s K
factor(s^3 + 10*s^2 + (21+K)*s + 4*K, s,'FactorMode','full')

ans = 

[rustem]

Actually at first i tried with s only. After that i tried with both. I forgot to undo the changes before i put it in here sorry for that and thanks all for the help.

[John D'Errico]

I think the problem is, even though factor seems like it should solve this, it looks for only more obvious factors. Can factor handle a simpler problem? Well, yes.

syms x
factor(x^2 + 4*x +4,x)

ans = 

How about this one? It handled that one.

syms K
factor(x^2 - 4*K*x + 4*K^2,x)

ans = 

But what if I give it something slightly less obvious?

factor(x^2 + 4*K^2 - 4*K,x)

ans = 

So factor was unable to solve even that one, even though we can write it simply enough in terms of sqrt(-4*K^2+4*K). Yet solve would handle it with not even a sweat.

[Dyuman Joshi]

I feel that the question should be - Why is factor() not able to do so?

Maybe I should raise a query to TMW about this.

[Torsten]

Maybe it factors only expressions with "nice" outcome :-)

[Walter Roberson]

Factorization of expressions with real components (or components of unknown reality) traditionally only proceeds to the point where any further factoring would require use of imaginary components.

syms x
factor(x^2 + 1)

ans = 

factor(x^2 - 1)

ans = 

P1 = expand((x-2) * (x+7) * (x-7))

P1 = 

factor(P1)

ans = 

P2 = expand((x-2) * (x+7i) * (x-7i))

P2 = 

factor(P2)

ans = 

P3 = expand((x-2i) * (x+7) * (x-7))

P3 = 

factor(P3)

ans = 

[Torsten]

I cannot see a rule when "factor" works and when if does not work.

syms x
P2 = expand((x-2) * (x+7i) * (x-7i))

P2 = 

factor(P2,x,'FactorMode','complex')

ans = 

Answer 2

[Torsten]

Here are the factors:

syms s K
eqn = s^3 + 10*s^2 + (21+K)*s + 4*K == 0;
factors = solve(eqn,s,'MaxDegree',3)

factors = 

[Dyuman Joshi]

Shouldn't the factors be

s - roots_of_equation

[Torsten]

Yes, I should have said: -s1, -s2 and -s3 from the factorization (s+s1)*(s+s2)*(s+s3).