Question

How do you factor and solve \[{x^3} - 216\]?

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 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.