Question

How do you factor ${x^6} - 64$?

Answer 1

Hint: Here we can proceed by writing $64$ as the cube of any number and also we can write ${x^6} = {\left( {{x^2}} \right)^3}$ and then we will get the given term as the form ${a^3} - {b^3}$
Now we can simplify it by using the formula:
${a^3} - {b^3} = (a - b)({a^2} + ab + {b^2})$

Complete step by step solution:
Here we are given to factorize the term ${x^6} - 64$ which means that we need to simplify it and write it in the form of the factors of it.
We know that a cube of any number is a product when it is multiplied by itself three times. For example: the cube of $1 = (1)(1)(1) = 1$
Cube of $2 = (2)(2)(2) = 8$
So we need to find the number whose cube is $64$ as we can simplify this equation in the general form ${a^3} - {b^3}$
So we know that cube of $4 = (4)(4)(4) = 64$
So we can write that $64 = {4^3}$
Also we can write ${x^6} = {\left( {{x^2}} \right)^3}$
Hence we can write the given term whose factor is to be found as:
${x^3} - 64$$ = {({x^2})^3} - {4^3}$
Now we know that we have the formula in the cube that when we have two cubes and we do their subtraction, we get:
$\Rightarrow$ ${a^3} - {b^3} = (a - b)({a^2} + ab + {b^2})$
So we can similarly compare ${x^6} - 64$ with ${a^3} - {b^3}$ and use the formula ${a^3} - $\Rightarrow$ {b^3} = (a - b)({a^2} + ab + {b^2})$
So we can compare and we get that as per the given equation:
$
  a = {x^2} \\
  b = 4 \\
 $
So we get that:
$\Rightarrow$ ${({x^2})^3} - {4^3} = ({x^2} - 4)({({x^2})^2} + 4{x^2} + {4^2})$
$\Rightarrow$ ${x^6} - 64 = ({x^2} - 4)({x^4} + 4{x^2} + 16)$
We can write ${x^2} - 4 = {x^2} - {2^2}$ and we will get:
$\Rightarrow$ ${x^6} - 64 = ({x^2} - {2^2})({x^4} + 4{x^2} + 16)$$ - - - - (1)$
Now we also know that we have the formula ${a^2} - {b^2} = (a + b)(a - b)$
So we can say that ${x^2} - {2^2} = (x + 2)(x - 2)$
Now we can put this value in equation (1) and we will get:
$\Rightarrow$ ${x^6} - 64 = (x + 2)(x - 2)({x^4} + 4{x^2} + 16)$
We can write $({x^4} + 4{x^2} + 16) = ({x^4} + 8{x^2} + 16 - 4{x^2})$
Now we can write it in the form $({x^4} + 4{x^2} + 16) = ({\left( {{x^2} + 4} \right)^2} - {\left( {2x} \right)^2})$
Now again we can apply the same formula ${a^2} - {b^2} = (a + b)(a - b)$and get the factors as:
$\Rightarrow$ ${x^6} - 64 = (x + 2)(x - 2)({x^2} + 2x + 4)({x^2} - 2x + 4)$
Hence we have found this simplified form of the factors of the given problem. We can simplify this again if we are able to get the proper factors of the further equation also. Here in the equation obtained that is $({x^2} + 2x + 4){\text{ and }}({x^2} - 2x + 4)$ we cannot simplify it further as discriminant is negative.

Note: Here in the simplified form obtained which is $(x + 2)(x - 2)({x^2} + 2x + 4)({x^2} - 2x + 4)$ we can also simplify the equation that is $({x^2} + 2x + 4){\text{ and }}({x^2} - 2x + 4)$ which means of the degree $2$ but its discriminant must be equal to zero.
Here it is negative as $D = {b^2} - 4ac{\text{ in }}a{x^2} + bx + c$ so here we have $D = {2^2} - 4(4) = 4 - 16 = {\text{negative}}$
So we don’t need to simplify it further.