Solve the Equation: x⁶ + x⁵ = 0 Using Common Factor Method

Shown below is the given equation:

$x^6+x^5=0$

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 $x^5$ given that the fifth power is the lowest power in the equation and therefore is included both in the term with the sixth power and in the term with the fifth power. Any power higher than this is not included in the term with the fifth power, which is the lowest, and therefore this is the term with the highest power that can be factored out as a common factor from all terms in the expression. Proceed with the factoring of the expression:

$x^6+x^5=0 \\ \downarrow\\ x^5(x+1)=0$

Let's continue 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 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:

$x^5=0 \hspace{8pt}\text{/}\sqrt[5]{\hspace{6pt}}\\ \boxed{x=0}$(in this case taking the odd root of the right side of the equation will yield one possibility)

or:

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

Let's summarize the solution of the equation:

$x^6+x^5=0 \\ \downarrow\\ x^5(x+1)=0 \\ \downarrow\\ x^5=0 \rightarrow\boxed{ x=0}\\ x+1=0 \rightarrow \boxed{x=-1}\\ \downarrow\\ \boxed{x=0,-1}$

Therefore the correct answer is answer A.