How to factorise $x^4$ equations?
I'm facing a problem to factorise this $64x^4+64x^3-88x^2-51x+39=0$. How to factorise $x^4$ equations?
$\endgroup$ 52 Answers
$\begingroup$If you can't find rational roots with the Rational Root Theorem, your next bet for solving this on paper is to factor the quartic into two quadratic equations. For example, in this case, we can let $$ 64x^4 + 64x^3 - 88x^2 - 51x + 39 = (ax^2 + bx + c)(px^2 + qx + r) $$ Note that, by comparing coefficients, we can tell that $ap = 64$, $cr = 39$, $br + cq = -51$, $aq + bp = 64$, $ar + bq + cp = -88$.
There are only a few values of $a,c, p, r$ that satisfy the first two equations. After some guessing and checking (kind of like the guessing and checking that goes into factoring a quadratic**), we find that $$ 64x^4 + 64x^3 - 88x^2 - 51x + 39 = (4x^2 + 3x - 3)(16x^2 + 4x - 13) $$
And you can finish by solving the two quadratics.
** Of course, this is a bit more involved than solving a quadratic. While there aren't many ways to simplify this process, you can make some educated guesses. For example, you can tell things about negative/positive, perhaps whether numbers are even/odd, etc. Things that make it easier.
$\endgroup$ $\begingroup$You can use this formula which always gives the solutions to quartic equations. Though the solutions won't be pretty...I get $$x_{1,2}=-\frac{1}{8}\pm \frac{\sqrt{53}}{8}, x_{3,4}=-\frac{3}{8}\pm\frac{\sqrt{57}}{8}.$$
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