Solve x⁷ - 5x⁶ = 0: Factoring High-Degree Polynomials

Shown below is the given problem:

$x^7-5x^6=0$

Note that on the left side we are able factor the expression using a common factor. The largest common factor for the numbers and letters in this case is $x^6$given that the sixth power is the lowest power in the equation. Therefore 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. Therefore this is the term with the highest power that can be factored out as a common factor from all terms in the expression, so we'll continue to factor the expression:

$x^7-5x^6=0 \\ \downarrow\\ x^6(x-5)=0$

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, since the only way to obtain 0 from a 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 due to the fact that we're dealing with zero, we only obtain one solution)

or:

$x-5=0\\ \downarrow\\ \boxed{x=5}$

Let's summarize the solution of the equation:

$x^7-5x^6=0 \\ \downarrow\\ x^6(x-5)=0 \\ \downarrow\\ x^6=0 \rightarrow\boxed{ x=0}\\ x-5=0 \rightarrow \boxed{x=5}\\ \downarrow\\ \boxed{x=0,5}$

Therefore the correct answer is answer A.