Solve the Polynomial Equation: 12x⁴ - 3x³ = 0

To solve this polynomial equation, we'll follow these steps:

• Step 1: Identify and factor out the greatest common factor (GCF).
• Step 2: Solve the resulting simpler equations for $x$.
• Step 3: Verify the solutions.

Now, let's work through each step:

Step 1: Factor out the Greatest Common Factor (GCF)

The given equation is $12x^4 - 3x^3 = 0$.
Both terms share a common factor of $3x^3$. Factoring out this common factor, we get:

$3x^3(4x - 1) = 0$

Step 2: Solve the factored equation

We now have two factors: $3x^3$ and $(4x - 1)$. Set each factor to zero to find possible solutions:

• For the factor $3x^3 = 0$:
  Solving this gives $x^3 = 0$, which implies $x = 0$.
• For the factor $4x - 1 = 0$:
  Solving this gives $4x = 1$, leading to $x = \frac{1}{4}$.

Step 3: Verification

Substitute $x = 0$ and $x = \frac{1}{4}$ back into the original equation to verify:

• Substituting $x = 0$:
  $12(0)^4 - 3(0)^3 = 0$, which is true.
• Substituting $x = \frac{1}{4}$:
  Calculations show $12\left(\frac{1}{4}\right)^4 - 3\left(\frac{1}{4}\right)^3 = 0$, which also holds true.

The correct choice from the given options is $x = 0, \frac{1}{4}$.

Therefore, the solution to the problem is $x=0,\frac{1}{4}$.