Question

How do you solve ${x^3} + {x^2} - 16x - 16 = 0$ ?

Answer 1

Hint: To solve this we have to know about algebraic identity.
Cubic equation: a polynomial equation in which the highest sum of exponents of a variable in any term is three. The value of $x$ for which the result of the equation is zero then that value will be the root of the cubic equation.

Complete step by step solution:
To solve this problem we should know about how to do factorization.
First we try and error method to calculate its root. Try to keep different values in place of $x$ . So, when the equation will give zero as output, that number will be the root of the cubic equation.
By observation we take $x = - 1$ . keeping it in equation:
 $ \Rightarrow {( - 1)^3} + {( - 1)^2} - 16( - 1) - 16 = - 1 + 1 + 16 - 16 = 0$
Hence, $ - 1$ is the one of the roots of a given equation.
So, one factor of the given equation will be;
 $(x - ( - 1)) = (x + 1)$
Divide given equation ${x^3} + {x^2} - 16x - 16$ by $(x + 1)$ . We get,
Divide it by using factorization method.
 ${x^3} + {x^2} - 16x - 16$
Take common ${x^2}$ from the first two terms and $16$ from last two.
 $ \Rightarrow {x^2}(x + 1) - 16(x + 1)$
Take $(x + 1)$ as common then we can write it as,
 $ \Rightarrow (x + 1)({x^2} - 16)$
So, divide $(x + 1)({x^2} - 16)$ by $(x + 1)$ . We get,
 $\dfrac{{(x + 1)({x^2} - 16)}}{{(x + 1)}} = {x^2} - 16$
So we can write it as,
 $ \Rightarrow {x^3} + {x^2} - 16x - 16 = (x + 1)\left( {{x^2} - 16} \right)$
By using algebraic identity. It write it as;
 $ \Rightarrow (x + 1)\left( {{x^2} - 16} \right) = (x + 1)\left( {x - 4} \right)\left( {x + 4} \right)$
  So, equating it with zero we get the root of the given cubic equation;
 $(x + 1)\left( {x - 4} \right)\left( {x + 4} \right) = 0$
By equating them separately we can write,
 $(x + 1) = 0$
 \[ \Rightarrow x = - 1\]
 $\left( {x - 4} \right)$
 \[ \Rightarrow x = 4\]
 $\left( {x + 4} \right)$
 \[ \Rightarrow x = - 4\]

Hence the root of the given cubic equation will be \[x = - 1, - 4\,and\,4\]

Note: Application of cubic equation is angle trisection and doubling the cube are two ancient problems of geometry that have been proved to not be solvable by straightedge and compass construction, because they are equivalent to solving a cubic equation.