Question

How do you factor \[{x^3} + 216\]?

Answer 1

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.