Originally Posted by

**iiharthero** Solve the equation 4xcubed - 12xsquared +9x - 2 =0 given that two of its roots are equal

$\displaystyle 4x^3 - 12x^2 + 9x - 2 = 0$

By the rational roots theorem, the possible rational roots are

$\displaystyle \frac{\text{factors of}\; -2}{\text{factors of}\; 4} = \frac{\pm 1, \pm 2}{\pm 1, \pm 2, \pm 4}$

= ±1, ±2, ±1/2, ±1/4.

Start testing these factors by using the Factor Theorem or synthetic division. You'll find that x = 2 is a root:

Code:

2| 4 -12 9 -2
-- 8 -8 2
-----------------
4 -4 1 0

So the polynomial factors into

$\displaystyle \begin{aligned}

4x^3 - 12x^2 + 9x - 2 &= (x - 2)(4x^2 - 4x + 1) \\

&= (x - 2)(2x - 1)(2x - 1)

\end{aligned}$

The other roots are x = 1/2 (double root).

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