Intereting Posts

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$$-8x^3 +8x -3 = 0$$

I’ve already tried the possible roots of $\pm 1$ and $3$ using the rational roots test, but none of these help break it down into something more workable. How do I solve this cubic without using online calculators?

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**Hint** $\ $ By the method below, we reduce to $\, -X^3\! +4X – 3 = 0\,$ for $\,X = 2x,\,$ whose only possible rational roots are $\,X= \pm1,\,\pm 3\,$ by the Rational Root Test. By inspection the coefficients sum to $0,\,$ i.e. $\,f(1) = 0,\,$ which yields the root $\,X = \,\ldots,\, $ thus $\ x = X/2 =\,\ldots$

**Key Idea** $\ \ $ Reduce to factoring a polynomial that is $\,\rm\color{#c00}{monic}\,$ (lead coeff $=1)$ as follows:

$$\quad\ \ \begin{eqnarray}

f &\,=\,& \ \ 9\ x^2-\ 80\ x\ -\,\ 9\\

\Rightarrow\ 9f &\,=\,& (9x)^2\! -80(9x)-81\\

&\,=\,& \ \ \ \ \color{#c00}{X^2\!- 80\ X\ -\,\ 81},\,\ \ X\, =\, 9x\\

&\,=\,& \ \ \ \,(X-81)\ (X+\,1)\\

&\,=\,& \ \ \ (9x-81)\,(9x+1)\\

\Rightarrow\ f\,=\, 9^{-1}(9f) &\,=\,& \ \ \ \ \ (x\ -\ 9)\,(9x+1)\\

\end{eqnarray}$$

If we denote our factoring algorithm by $\,\cal F,\,$ then the above transformation is simply

$$\cal F f\, = a^{-1}\cal F\, a\,f\quad\,$$

Thus we’ve transformed by $ $ *conjugation* $\,\ \cal F = a^{-1} \cal F\, a\ \,$ the problem of factoring non-monic polynomials into the simpler problem of factoring monic polynomials.

This is sometimes called the **AC method**. It works for higher degree polynomials too. As above, we can reduce the problem of factoring a non-monic polynomial to that of factoring a monic polynomial by scaling by a $ $ *power* of the lead coefficient $\rm\:a\:$ then changing variables: $\rm\ X = a\:x$

$$\begin{eqnarray} \rm\: a\:f(x)\:\! \,=\,\:\! a\:(a\:x^2 + b\:x + c) &\,=\,&\rm\: X^2 + b\:X + \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\smash[t]{\overbrace{ac}^{\rm\qquad\ \ \ \ \ {\bf AC-method}}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! =\, g(X),\ \ \ X = a\:x \\

\\

\rm\: a^{n-1}(a\:x^n\! + b\:x^{n-1}\!+\cdots+d\:x + c) &\,=\,&\rm\: X^n\! + b\:X^{n-1}\!+\cdots+a^{n-2}d\:X + a^{n-1}c

\end{eqnarray}$$

After factoring the monic $\rm\,g(X)\, =\, a^{n-1}f(x),\,$ we are guaranteed that the transformation reverses to yield a factorization of $\rm\:f,\ $ since $\rm\ a^{n-1}$ must divide into the factors of $\rm\ g\ $ by Gauss’ Lemma, i.e. primes $\,p\in\rm\mathbb Z\,$ remain prime in $\rm\,\mathbb Z[X],\,$ so $\rm\ p\ |\ g_1(x)\:g_2(x)\,$ $\Rightarrow$ $\,\rm\:p\:|\:g_1(x)\:$ or $\rm\:p\:|\:g_2(x).$

This method also works for multivariate polynomial factorization, e.g. it applies to this question.

**Remark** $\ $ Those who know university algebra might be interested to know that this works not only for UFDs and GCD domains but also for integrally-closed domains satisfying

$\qquad\qquad$ Primal Divisor Property $\rm\ \ c\ |\ AB\ \ \Rightarrow\ \ c = ab,\ \ a\ |\: A,\ \ b\ |\ B$

Elements $c$ satisfying this are called *primal.* One easily checks that atoms are primal $\!\iff\!$ prime. Also products of primes are also primal. So “primal” may be viewed as a generalization of the notion “prime” from atoms (irreducibles) to composites.

Integrally closed domains whose elements are all primal are called $ $ *Schreier rings* by Paul Cohn (or *Riesz domains*, because they satisfy a divisibility form of the Riesz interpolation property). In Cohn’s Bezout rings and their subrings

he proved that if $\rm\:D\:$ is Shcreier then so too is $\rm\,D[x],\:$ by using a primal analogue of Nagata’s Lemma: an atomic domain $\rm\:D\:$ is a UFD if some localization $\rm\:D_S\:$ is a UFD, for some monoid $\rm\:S\:$ generated by primes. These primal and Riesz interpolation viewpoints come to the fore in a *refinement view* of unique factorization, which proves especially fruitful in noncommutative rings (e.g. see Cohn’s 1973 *Monthly* survey Unique factorization domains).

In fact Schreier domains can be characterized equivalently by a suitably formulated version of the above “factoring by conjugation” property. This connection between this elementary AC method and Schreier domains appears to have gone unnoticed in the literature.

Note that your polynomial is *not* monic but has leading coefficient $-8$. Thus, by the rational root theorem, you have to test

$$

\pm 1, \pm 3, \pm \frac 1 2, \pm \frac 3 2, \pm \frac 1 4, \pm \frac 3 4, \pm \frac 1 8, \text{ and } \pm \frac 3 8

$$

to find all rational roots of $-8x^3 + 8x – 3 = 0$.

One of these will indeed be a root and long division will leave you with a quadratic equation for the other two roots.

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