Question

How do you factor and solve ${x^3} - 216$?

Answer 1

Here in this question, we have to find the factors of the given equation. If you see the equation it is in the form of ${a^3} - {b^3}$. We have a standard formula on this algebraic equation and it is given by ${a^3} - {b^3} = (a - b)({a^2} + ab + {b^2})$, hence by substituting the value of a and b we find the factors.

Complete step-by-step solution:
The equation is an algebraic equation or expression, where algebraic expression is a combination of variables and constant. nNow consider the given equation ${x^3} - 216$, let we write in the exponential form. The number ${x^3}$ can be written as $x \times x \times x$ and the $216$can be written as $6 \times 6 \times 6$, in the exponential form it is ${\left( 6 \right)^3}$. The number ${x^3}$ written as $x \times x \times x$ and in exponential form is ${\left( x \right)^3}$. Therefore, the given equation is written as ${\left( x \right)^3} - {\left( 6 \right)^3}$, the equation is in the form of ${a^3} - {b^3}$. We have a standard formula on this algebraic equation and it is given by ${a^3} - {b^3} = (a - b)({a^2} + ab + {b^2})$, here the value of a is $x$ and the value of b is $6$.

By substituting these values in the formula, we have
${x^3} - 216 = {\left( x \right)^3} - {\left( 6 \right)^3} = (x - 6)({(x)^2} + (x)(6) + {(6)^2})$
On simplifying we have
$\Rightarrow {x^3} - 216 = (x - 6)({x^2} + 6x + 36)$
The second term of the above equation can be solved further by using factorisation or by using the formula $({a^2} + {b^2}) = {(a + b)^2} - 2ab$
The above equation is written as
$\Rightarrow {x^3} - 216 = (x - 6)({x^2} + 36 + 6x)$
So let we consider the second term and solve it so we have
$\Rightarrow ({x^2} + 36 + 6x) = {(x + 6)^2} - 2(x)(6) + 6x$
On simplifying we have
$\Rightarrow ({x^2} + 36 + 6x) = {(x + 6)^2} - 12x + 6x$
On further simplification we have
$\Rightarrow ({x^2} + 36 + 6x) = {(x + 6)^2} - 6x$
If we see the simplification of the second term, it looks like the bilk term. So there is no need to simplify the second term.

Therefore, the factors of ${x^3} - 216$ is $(x - 6)({x^2} + 6x + 36)$

Note: To find the factors for algebraic equations or expressions, it depends on the degree of the equation. If the equation contains a square then we have two factors. If the equation contains a cube then we have three factors. Here this equation also contains 3 factors, the two factors may be imaginary.