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

Shown below is the given equation:

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

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 variables in this case is $7x^7$due to the fact that the seventh power is the lowest power in the equation. Therefore it 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. Hence 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 to factor the expression:

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

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, given 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:

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

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.