Solving Equation

• August 7th 2010, 03:23 AM
winsome
Solving Equation
solv for x, 6x^3 -7x^2 -12x +7 = 0
• August 7th 2010, 05:58 AM
eumyang
Using the Rational Zeros Test:
$\dfrac{Factors\:of\:7}{Factors\:of\:6} = \dfrac{\pm 1,\,\pm 7}{\pm 1,\,\pm 2,\,\pm 3,\,\pm 6}$

... and guess-and-check with these potential rational zeros, I see that 1/2 is a zero. So the polynomial factors to:
$6x^3 - 7x^2 - 12x + 7 = (2x - 1)(3x^2 - 2x - 7)$

The quadratic isn't factorable, so you'll have to use the quadratic formula to get the other two roots.
• August 7th 2010, 06:35 AM
Prove It
Isn't it?

$(2x - 1)(3x^2 -2x - 7) = 3(2x -1)\left(x^2 - \frac{2}{3}x - \frac{7}{3}\right)$

$= 3(2x - 1)[x^2 - \frac{2}{3}x + \left(-\frac{1}{3}\right)^2 - \left(-\frac{1}{3}\right)^2 - \frac{7}{3}\right]$

$= 3(2x - 1)\left[\left(x-\frac{1}{3}\right)^2-\frac{22}{9}\right]$

$= 3(2x-1)\left(x-\frac{1}{3}+\frac{\sqrt{22}}{3}\right)\left(x-\frac{1}{3}-\frac{\sqrt{22}}{3}\right)$.
• August 7th 2010, 06:46 AM
eumyang
Touché. (Giggle) (Of course I meant factorable over the rationals. (Wink))