Question

How do I factor \[ {x^3} - 2{x^2} - 4x + 8\] completely?

Answer 1

Hint: Here, we will factorize the given polynomial by grouping. We will group the first two terms and the last two terms of the given polynomial together. Then taking the factors common from each pair, we will factor them in the form of a linear expression and a quadratic expression. We will then factorize the quadratic expression using a suitable algebraic identity to get the required value.

Complete step-by-step solution:
Let us write the polynomial to be factored given in the question as
\[p\left( x \right) = {x^3} - 2{x^2} - 4x + 8\]
Now, we make the two pairs of the above polynomial by combining the first two terms and the last two terms together. Therefore, we get
\[ \Rightarrow p\left( x \right) = \left( {{x^3} - 2{x^2}} \right) + \left( { - 4x + 8} \right)\]
Now, we can see that \[{x^2}\] is common to both the terms in the first pair, and \[ - 4\] is common to both the terms in the second pair.
Taking these common from the respective pairs, we get
\[ \Rightarrow p\left( x \right) = {x^2}\left( {x - 2} \right) - 4\left( {x - 2} \right)\]
Now, we can take \[\left( {x - 2} \right)\] common from both the terms in the above equation to get
\[ \Rightarrow p\left( x \right) = \left( {x - 2} \right)\left( {{x^2} - 4} \right)\]
Now we can write 4 in the above equation as\[2^2\]. Therefore, we get
\[ \Rightarrow p\left( x \right) = \left( {x - 2} \right)\left( {{x^2} - {2^2}} \right)\]
Now, we know that \[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]. So the above equation can be written as
\[ \Rightarrow p\left( x \right) = \left( {x - 2} \right)\left( {x + 2} \right)\left( {x - 2} \right)\]
\[ \Rightarrow p\left( x \right) = {\left( {x - 2} \right)^2}\left( {x + 2} \right)\]
Hence, the given polynomial is factored in completely.

Note:
The given polynomial is a cubic polynomial. A cubic polynomial is a polynomial that has the highest degree of variable as 3. Similarly, a quadratic polynomial is a polynomial that has the highest degree of variable as 2. We can say that we have factored the expression completely when we use the form of linear expression. That means if we have a quadratic expression we have to factorize it to get a linear expression, then only we can say that the expression is factored completely.