Question

How do you factor ${x^3} + 8?$

Answer 1

Hint: The given problem is to find the factor of ${x^3} + 8$, so find the factor we can rewrite the given problem as ${x^3} + {2^3}$. Now this is in the form of ${a^3} + {b^3}$ . By using the formula ${a^3} + {b^3} = (a + b)({a^2} - ab + {b^2})$ we can easily find the factor of the given problem.

Complete step-by-step answer:
In the given question they have asked to find the factor of ${x^3} + 8$. To find the factor first try to get the form so that we can apply any formula to reduce that.
We have a formula for ${a^3} + {b^3}$ as ${a^3} + {b^3} = (a + b)({a^2} - ab + {b^2})$ , here $a$ can be replaced by $x$ and to write $b$ we should rewrite the given problem statement.
So, we can rewrite the ${x^3} + 8$ as ${x^3} + {2^3}$, therefore the value of $b$ becomes $2$ .
By substituting the value of $a$ as $x$ and value of $b$ as $2$ in the ${a^3} + {b^3}$ equation, we get
\[{x^3} + {2^3} = (x + 2)({x^2} - 2x + {2^2})\]
\[(x + 2)\] and \[({x^2} - 2x + {2^2})\] are the factors of ${x^3} + 8$.
To reduce or simplify further we can use the quadratic equation formula to solve.
\[({x^2} - 2x + 4)\]. This is in the form of a quadratic equation, so we use the quadratic equation formula to solve.
Quadratic formula for standard form $a{x^2} + bx + c$ is given by:
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4(a)(c)} }}{{2(a)}}$
Here, $a = 1$ , $b = - 2$ and $c = 4$, by substituting these in the above equation, we get
$x = \dfrac{{ - ( - 2) \pm \sqrt {{{( - 2)}^2} - 4(1)(4)} }}{{2(1)}}$
On simplification, we get
$ \Rightarrow x = \dfrac{{2 \pm \sqrt {4 - 16} }}{2}$
$ \Rightarrow x = \dfrac{{2 \pm \sqrt { - 12} }}{2}$
$ \Rightarrow x = \dfrac{{2 \pm \sqrt { - (3 \times 4)} }}{2}$
$ \Rightarrow x = \dfrac{{2 \pm 2i\sqrt 3 }}{2}$ (here, we have taken -2 outside so – is represented in terms of i )
$ \Rightarrow x = 1 \pm 1i\sqrt 3 $
The factors are $x - 1 + 1i\sqrt 3 $ and $x - 1 - 1i\sqrt 3 $
Therefore the factors of ${x^3} + 8$ are \[(x + 2)\] , $x - 1 + 1i\sqrt 3 $ and $x - 1 - 1i\sqrt 3 $ .

Note: Whenever we are finding a factor of any given values first try to reduce to any equation form so that we can apply any equation and solve to get the required answer. If we get a quadratic equation while solving them you can apply a quadratic formula but carefully you need to simplify as it involves imaginary roots.