Solve the Polynomial Equation: 3x³ - 10x² + 7x = 0 for X

To solve the problem $3x^3 - 10x^2 + 7x = 0$, follow these steps:

• Step 1: Factor out the greatest common factor (GCF). The common factor in all terms is $x$. Factoring out $x$ gives:

$x(3x^2 - 10x + 7) = 0$

• Step 2: Apply the Zero Product Property. Set each factor equal to zero:

$x = 0$ or $3x^2 - 10x + 7 = 0$

• Step 3: Solve the quadratic equation $3x^2 - 10x + 7 = 0$ using the quadratic formula:

The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Here, $a = 3$, $b = -10$, $c = 7$.

Calculate the discriminant:

$b^2 - 4ac = (-10)^2 - 4(3)(7) = 100 - 84 = 16$

Since the discriminant is a perfect square, this quadratic has rational roots. Using the quadratic formula gives:

$x = \frac{-(-10) \pm \sqrt{16}}{6} = \frac{10 \pm 4}{6}$

Thus, the solutions are:

$x = \frac{10 + 4}{6} = \frac{14}{6} = \frac{7}{3}$

$x = \frac{10 - 4}{6} = \frac{6}{6} = 1$

• Step 4: Combine all solutions. The solutions to the original equation are:

$x = 0$, $x = 1$, and $x = \frac{7}{3}$.

Therefore, the solution to the problem is $x = 0, 1, \frac{7}{3}$.