Solve the Polynomial Equation: 15x⁴ - 30x³ = 0

Shown below is the given equation:

$15x^4-30x^3=0$

First, note that on the left side we are able to factor the expression by using a common factor. The largest common factor for the numbers and variables in this case is $15x^3$given that the third power is the lowest power in the equation. Therefore it is included in both the term with the fourth power and the term with the third power. Any power higher than this is not included in the term with the third power, which is the lowest. Hence this is the term with the highest power that can be factored out as a common factor from all terms for the variables,

For the numbers, note that 30 is a multiple of 15, therefore 15 is the largest common factor for the numbers for both terms in the expression,

Let's continue to factor the expression:

$15x^4-30x^3=0 \\ \downarrow\\ 15x^3(x-2)=0$

Proceed to the left side of the equation that we obtained in the last step. There is a multiplication of algebraic expressions and on the right side the number 0. Therefore due to the fact that the only way to obtain 0 as a result of multiplication is to multiply by 0, at least one of the expressions in the multiplication on the left side must equal zero,

Meaning:

$15x^3=0 \hspace{8pt}\text{/}:15\\ x^3=0 \hspace{8pt}\text{/}\sqrt[3]{\hspace{6pt}}\\ \boxed{x=0}$

In solving the equation above, we first divided both sides of the equation by the term with the unknown and then proceeded to extract a cube root for both sides of the equation.

(In this case extracting an odd root for the right side of the equation yielded one possibility)

Or:

$x-2=0\\ \boxed{x=2}$

Let's summarize the solution of the equation:

$15x^4-30x^3=0 \\ \downarrow\\ 15x^3(x-2)=0 \\ \downarrow\\ 15x^3=0 \rightarrow\boxed{ x=0}\\ x-2=0\rightarrow \boxed{x=2}\\ \downarrow\\ \boxed{x=0,2}$

Therefore the correct answer is answer B.