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# How do you factor ${x^3} + 216$?

Last updated date: 19th Jun 2024
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Hint: 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 constants.

Now 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 ${x^3}$ is written as $x \times x \times x$ and in exponential form is ${(x)^3}$. Therefore, the given equation is written as ${\left( x \right)^3} + {6^3}$, the equation is in the form of ${a^3} + {b^3}$.The ${a^3} + {b^3}$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
${x^3} + 216 = {\left( x \right)^3} + {6^3} = (x + 6)({(x)^2} - (x)(6) + {6^2})$
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 $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Let we consider ${x^2} - 6x + 36$, and find factors for this. Here a=1, b=-6 and c=36. By substituting these values in the formula we get
$x = \dfrac{{ - ( - 6) \pm \sqrt {{{( - 6)}^2} - 4(1)(16)} }}{{2(1)}}$
On simplification we have
$\Rightarrow x = \dfrac{{6 \pm \sqrt {36 - 64} }}{2}$
$\Rightarrow x = \dfrac{{6 \pm \sqrt { - 28} }}{2}$
On further simplifying we get an imaginary number so let us keep as it is.
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 are imaginary.