Question

Solve \[{(x - 2)^3}\].

Answer 1

Hint: Here we substitute the values of \[a = x\] and \[b = 2\] in the formula of \[{(a - b)^3}\] and solve the expansion using basic addition, subtraction and multiplication. Always add the terms having the same variable and write the equation in descending order of power of the variable. Example \[{x^6} + 5{x^4} + 3x + 1 = 0\] is the correct way to write an equation.
* Any number having power \[n\] can be written as \[n\] times multiple of itself, I.e. \[{a^n} = \{ \underbrace {a \times a \times .... \times a}_n\} \]

Complete step-by-step answer:
To solve \[{(x - 2)^3}\] we first compare it to \[{(a - b)^3}\]
So, \[a = x\] and \[b = 2\]
Since, we know \[{(a - b)^3} = {a^3} - {b^3} - 3ab(a - b)\] \[...(i)\]
Therefore substituting the values of \[a = x\] and \[b = 2\] in equation \[(i)\].
\[{(x - 2)^3} = {(x)^3} - {(2)^3} - 3(x)(2)(x - 2)\]
Using the expansion of power \[{a^n} = \{ \underbrace {a \times a \times .... \times a}_n\} \] write \[{2^3} = 2 \times 2 \times 2\]
               \[
   = {x^3} - (2 \times 2 \times 2) - x(3 \times 2)(x - 2) \\
    \\
 \]
                \[
   = {x^3} - (8) - 6x(x - 2) \\
    \\
 \]
               \[
   = {x^3} - (8) - \{ 6x \times (x) + 6x \times ( - 2)\} \\
    \\
 \]
               \[
   = {x^3} - 8 - (6{x^2} - 12x) \\
    \\
 \]
               \[ = {x^3} - 8 - 6{x^2} + 12x\]
Writing the terms in increasing order of powers of variable \[x\]
               \[ = {x^3} - 6{x^2} + 12x - 8\]
Therefore, the value of \[{(x - 2)^3} = {x^3} - 6{x^2} + 12x - 8\]

Additional information:
Some other common expansions useful for these type of questions are
1) \[{(a - b)^2} = {a^2} + {b^2} - 2ab\]
2) \[{(a + b)^2} = {a^2} + {b^2} + 2ab\]
3) \[{(a + b)^3} = {a^3} + {b^3} + 3ab(a + b)\]

Note:
These types of questions can be solved by various methods. Since it includes expansion and we can expand directly, or by breaking down into lower terms therefore there are more than one ways to solve this.
Alternative method:
This method involves direct breaking up of the power into multiplication i.e. \[{a^n} = \{ \underbrace {a \times a \times .... \times a}_n\} \]
We can write \[{(x - 2)^3} = (x - 2) \times (x - 2) \times (x - 2)\]
Therefore multiplying the brackets on the right hand side one by one.
Firstly, we multiply the first two brackets and keep the third bracket separately outside.
\[{(x - 2)^3} = \{ (x - 2) \times (x - 2)\} \times (x - 2)\]
               \[ = \{ x \times (x - 2) - 2 \times (x - 2)\} \times (x - 2)\]
               \[ = \{ (x \times x) - (2 \times x) - 2 \times (x) - 2 \times ( - 2)\} \times (x - 2)\]
               \[ = \{ {x^2} - 2x - 2x + 4\} \times (x - 2)\]
               \[ = \{ {x^2} - 4x + 4\} \times (x - 2)\]
Now, we multiply the product obtained with the third bracket.
                \[ = \{ {x^2}(x - 2) - 4x(x - 2) + 4(x - 2)\} \]
                \[ = \{ {x^2} \times (x) - {x^2} \times (2) - 4x \times (x) - 4x \times ( - 2) + 4 \times (x) + 4 \times ( - 2)\} \]
                \[ = \{ {x^3} - 2{x^2} - 4{x^2} + 8x + 4x - 8\} \]
                \[ = \{ {x^3} - 6{x^2} + 12x - 8\} \]
Alternate method:
Since, we know the property of exponents that powers can be added when the base is the same.
Therefore, we can write the power \[3 = 2 + 1\] which gives us
\[{(x - 2)^3} = {(x - 2)^{2 + 1}}\]
Now we can separate the terms on the right side of the equation as powers are being added.
\[{(x - 2)^3} = {(x - 2)^2} \times (x - 2)\]
First we solve \[{(x - 2)^2}\] using the formula \[{(a - b)^2} = {a^2} + {b^2} - 2ab\]
Substitute value of \[a = x\] and \[b = 2\]