Solve the Polynomial Equation: 7x^8 - 21x^7 = 0 Using Common Factors

The equation in the problem is:

$7x^8-21x^7=0$

First, let's note that in the left side we can factor the expression using a common factor, the largest common factor for the numbers and variables in this case is $7x^7$since the seventh power is the lowest power in the equation and therefore is included in both the term with the eighth power and the term with the seventh power. Any power higher than this is not included in the term with the seventh power, which is the lowest, and therefore this is the term with the highest power that can be factored out as a common factor for variables,

For the numbers, we notice that 21 is a multiple of 7, therefore 7 is the largest common factor for numbers in both terms of the expression,

Let's continue and perform the factoring:

$7x^8-21x^7=0 \\ \downarrow\\ 7x^7(x-3)=0$

Let's continue and address the fact that in the left side of the equation we obtained in the last step there is a multiplication of algebraic expressions and on the right side the number 0, therefore, since the only way to get 0 from multiplication is to multiply by 0, at least one of the expressions in the multiplication on the left side must equal zero,

Meaning:

$7x^7=0 \hspace{8pt}\text{/}:7\\ x^7=0 \hspace{8pt}\text{/}\sqrt[7]{\hspace{6pt}}\\ \boxed{x=0}$

In solving the equation above, we first divided both sides of the equation by the term with the variable and then extracted a seventh root from both sides of the equation.

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

Or:

$x-3=0\\ \boxed{x=3}$

Let's summarize the solution of the equation:

$7x^8-21x^7=0 \\ \downarrow\\ 7x^7(x-3)=0 \\ \downarrow\\ 7x^7=0 \rightarrow\boxed{ x=0}\\ x-3=0\rightarrow \boxed{x=3}\\ \downarrow\\ \boxed{x=0,3}$

Therefore, the correct answer is answer B.