Question

How do you factor completely $ 125{x^3} + 1 $ ?

Answer 1

Hint: In order to factor out the above cubic expression, rewrite the coefficient of the cubic term as $ {5^3} $ and apply the formula of sum of cube of two terms $ {A^3} + {B^3} = (A + B)({A^2} - A.B + {B^2}) $ by considering the $ A\, $ as $ 5x $ and $ B $ as $ 1 $ ,you will get your required factors.
Formula Used:
 $ {A^3} + {B^3} = (A + B)({A^2} - A.B + {B^2}) $

Complete step-by-step answer:
We are given a cubic expression having variable $ x $ i.e. $ 125{x^3} + 1 $
In order to factorise the above cubic expression completely, we will rewrite the expression
Since, $ 125 $ can be written as the cube of the number $ 5 $ , $ 125 = 5 \times 5 \times 5 = {5^3} $ .
Putting this into the original expression, we get
 $
   \Rightarrow 125{x^3} + 1 \\
   \Rightarrow {5^3}{x^3} + 1 \\
   \Rightarrow {\left( {5x} \right)^3} + {\left( 1 \right)^3} \;
  $
Now applying the formula of sum of cube of two numbers as $ {A^3} + {B^3} = (A + B)({A^2} - A.B + {B^2}) $ by considering $ A\, $ as $ 5x $ and $ B $ as $ 1 $ .
 $
   \Rightarrow \left( {5x + 1} \right)\left( {{{\left( {5x} \right)}^2} - \left( {5x} \right)\left( 1 \right) + {{\left( 1 \right)}^2}} \right) \\
   \Rightarrow \left( {5x + 1} \right)\left( {25{x^2} - 5x + 1} \right) \;
  $
Therefore, the cubic expression $ 125{x^3} + 1 $ in the factor form is $ \left( {5x + 1} \right)\left( {25{x^2} - 5x + 1} \right) $
So, the correct answer is “ $ \left( {5x + 1} \right)\left( {25{x^2} - 5x + 1} \right) $ ”.

Note: Cubic Equation: A cubic equation is a equation which can be represented in the form of $ a{x^3} + b{x^2} + cx + d = 0 $ where $ x $ is the unknown variable and a,b,c,d are the numbers known where $ a \ne 0 $ .If $ a = 0 $ then the equation will become quadratic equation and will no more cubic .
Every cubic equation has 3 roots.
If you draw the graph of any cubic equation ,the points at which the curve intersects the x-axis are the values at which the equation has its solutions or we can say they are the root to the equation.
The value of $ 1 $ raised to power n, where n is any number is always equal to 1.Writing factors of any expression is actually writing the expression in the form of one term that is in the form of product $ \left( {} \right)\left( {} \right)\left( {} \right)... $