How do you factor $2{x^3} - 3 = 125$ ?

Hint: The degree of a polynomial equation is defined as the highest exponent of the unknown quantity in a polynomial equation. The factors/solution/zeros of the given polynomial are defined as those values of x at which the value of a polynomial is zero. In the given question, we are given a polynomial equation whose degree is 3 and we have to factor the given equation, so for that, we will first simplify the equation and then find one of its factors either by hit and trial or by some arithmetic identity.

Complete step-by-step solution:
We are given that $2{x^3} - 3 = 125$ and we have to factor this equation, so will rearrange this equation as –
$
  2{x^3} - 3 = 125 \\
   \Rightarrow 2{x^3} - 3 - 125 = 0 \\
   \Rightarrow 2{x^3} - 128 = 0 \\
   \Rightarrow 2({x^3} - 64) = 0 \\
   \Rightarrow {x^3} - 64 = 0 \\
 $
Now, we know that
\[{a^3} - {b^3} = (a - b)({a^2} + {b^2} + ab)\]
So, we get –
$
  {x^3} - {(4)^3} = 0 \\
   \Rightarrow (x - 4)({x^2} + {(4)^2} + x \times 4) = 0 \\
   \Rightarrow (x - 4)({x^2} + 16 + 4x) = 0 \\
 $
Now, ${x^2} + 16 + 4x$ cannot be factorized, so we solve it by using the quadratic formula –
$
  x = \dfrac{{ - 4 \pm \sqrt {{4^2} - 4(1)(16)} }}{{2(1)}} \\
   \Rightarrow x = \dfrac{{ - 4 \pm \sqrt { - 3(16)} }}{2} = \dfrac{{ - 4 \pm 4\sqrt 3 i}}{2} \\
   \Rightarrow x = - 2 \pm 2\sqrt 3 i \\
 $
Hence, the factors of the equation, $2{x^3} - 3 = 125$ are $x - 4 = 0$ , $x + 2 - \sqrt 3 i = 0$ and $x + 2 + \sqrt 3 i = 0$ .

Note: An expression that contains numerical values and alphabets too is known as an algebraic expression, but when the alphabets representing an unknown variable quantity are raised to some non-negative integer as the power, the algebraic expression becomes a polynomial equation. A quadratic polynomial is defined as a polynomial of degree two and its zeros can be found using many methods like factorization, completing the square, graphs, quadratic formula etc. We use the quadratic formula when we fail to find the factors of the equation. For applying the quadratic formula, we first express the given equation in the standard equation form and then the values of the coefficients are plugged in the quadratic formula.