Factorization of quartic polynomial.

Question

I want to know other ways of factorization to get quadratic factors in this polynomial:

$$x^4+2x^3+3x^2+2x-3$$

Thanks in advance for your suggestions.

The original polynomial is $$x^6-2x^3-4x^2+8x-3$$ where the two found factors are $(x+1)$ and $(x-1)$ by synthetic division.

Answer 1

Note that $$(x^2 + x + 1)^2 = x^4 + 2x^3 + 3x^2 + 2x + 1$$which means that your polynomial is equal to $$ (x^2 + x + 1)^2 - 4 \\ = (x^2 + x + 1)^2 - 2^2\\ = (x^2 + x + 1 - 2)(x^2 + x + 1 + 2) $$

Answer 2

One may try to use the rational root theorem. But that is not possible in this case because there is no rational root.

Another thing we can try (if we don't "see" anything) is write

$$(x^2+ax+b)(x^2+cx+d)\equiv x^4+2x^3+3x^2+2x-3$$

$$x^4+(a+c)x^3+(d+ac+b)x^2+(ad+bc)x+bd\equiv x^4+2x^3+3x^2+2x-3$$

and then solve:

\begin{cases} a+c=2\\ d+ac+b=3\\ ad+bc=2\\ bd=-3 \end{cases}

EDIT:

Once this answer is getting attention I will explain more.

Like already said, that is one approach when we don't have any good idea about how to proceed. Solve the system depends how it looks and each system is a new system. However that idea works for many cases that we usually face with.

The first idea to solve the system is look for integer solutions. On that way, the best equation to start is $bd=-3$. That give us the possibilities $(b,d)=\{(-1,3),(1,-3),(-3,1),(3,-1)\}$. Now we plug those possibilities in the system and check if we find $a$ and $c$ that solve it. After do that we see that the solutions are $(a,b,c,d)=\{(1,-1,1,3),(1,3,1,-1)\}$. But both give us the same factorization.

Answer 3

Rational Root Theorem Solution

We can reduce the problem to a straightforward Rational Root Theorem problem as follows. We start with reducing the polynomial modulo $x^2 - p x - q$ where $p$ and $q$ are undetermined integers that we want to choose such that this yields zero. This amounts to putting $x^2 = p x + q$ and higher powers are reduced by multiplying this rule by $x$ and then applying this reduction rule again. So, the polynomial becomes a linear function by applying the substitutions:

$$\begin{split} x^2 &= p x + q\\ x^3 &= p x^2 + q x&= (p^2 + q)x + pq\\ x^4 &= p x^3 + q x^2 &= p(p^2+2q)x+q(p^2+q) \end{split}$$

The polynomial reduced modulo $x^2 - p x - q$ then becomes:

$$\left(p^3 + 2 pq + 2 p^2 +2 q+3p + 2\right)x +p^2 q + q^2 + 2 pq + 2 q-3\tag{1}$$

We need to find integers $p$ and $q$ such that the above function is identical zero. We can simplify things by using that the function is then also zero for special values for $x$, allowing us to choose a value for $x$ that simplifies things a lot. Choosing $x = -\frac{q}{p}$ eliminates the $p^3$ term and a few more terms, setting what remains equal to zero yields a linear equation for $p$ in terms of $q$. Expressing $p$ in terms of $q$ yields:

$$p = -\frac{2q(q+1)}{q^2+q+3}\tag{2}$$

We can then substitute this in the constant term of eq. (1), setting the resulting expression equal to zero yields the equation:

$$q^6 + 3 q^5 + 7 q^4 + 10 q^3 - 21 q^2 + 27 q - 27 = 0\tag{3}$$

Here we have set the numerator of the resulting rational function equal to zero. Now we know that $q$ must be an integer, therefore the Rational Root Theorem is guaranteed to succeed and inserting any solution for $q$ (for which denominator is not zero) in eq. (2) will always yield a $p$ such that $x^2 - p x - q$ is a factor of the polynomial. It can be easily shown that this method works for general fourth degree polynomials.

Note also that applying the rational root theorem to find integer solution can be made very efficient by shifting the variable. If the value of a polynomial for some value $u$ is not zero, then this is the value of the constant term of the polynomial obtained by translating the variable by $u$. So, by factoring the value you just obtained you obtain the root candidates of that translated polynomial, adding $u$ to those values yields the root candidates of the original polynomial. The root candidates must then be on the intersection of the original list and the new list. By repeating this method, the list of root candidates will shrink rapidly. This is why I find the Rational Root Theorem such a powerful tool and that's why at least in my opinion it's worthwhile to reduce problems to Rational Root Theorem problems.