Solve the Polynomial Equation: x^7 - x^6 = 0 Using Factoring

Shown below is the given problem:

$x^7-x^6=0$

First, note that on the left side we are able factor the expression by using a common factor.

The largest common factor for the numbers and letters in this case is $x^6$due to the fact that the sixth power is the lowest power in the equation . Therefore it is included both in the term with the seventh power and in the term with the sixth power. Any power higher than this is not included in the term with the sixth 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 in the expression. Continue to factor the expression.

$x^7-x^6=0 \\ \downarrow\\ x^6(x-1)=0$

Proceed to the left side of the equation that we obtained from 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 from multiplication is to multiply by 0, at least one of the expressions in the multiplication on the left side must equal zero,

Meaning:

$x^6=0 \hspace{8pt}\text{/}\sqrt[6]{\hspace{6pt}}\\ x=\pm0\\ \boxed{x=0}$

(in this case taking the even root of the right side of the equation will yield two possibilities - positive and negative however given that we're dealing with zero, we only obtain one answer)

or:

$x-1=0\\ \downarrow\\ \boxed{x=1}$

Let's summarize the solution of the equation:

$x^7-x^6=0 \\ \downarrow\\ x^6(x-1)=0 \\ \downarrow\\ x^6=0 \rightarrow\boxed{ x=0}\\ x-1=0 \rightarrow \boxed{x=1}\\ \downarrow\\ \boxed{x=0,1}$

Therefore the correct answer is answer C.