Question

Factorise the following expression: $\left( {{x^2} - 4x} \right)\left( {{x^2} - 4x - 1} \right) - 20$

Answer 1

Hint: To solve this question, we will use the concept of factorisation of polynomials. The factorisation is a process of braking of a polynomial into the product of its factors. If $p\left( x \right)$ is a polynomial of degree $n \geqslant 1$ and a is any real number, then (i) x – a is a factor of $p\left( x \right)$, if $p\left( a \right) = 0$, and (ii) $p\left( a \right) = 0$, if x – a is a factor of $p\left( x \right)$.

Complete step-by-step answer:
An expression consisting of variables ( that are also known as indeterminates ) and coefficients that involve only the addition, subtraction, multiplication and non-negative integer exponentiation operations of variables is called a polynomial.
Given,
$\left( {{x^2} - 4x} \right)\left( {{x^2} - 4x - 1} \right) - 20$ ……….. (i)
Let us assume $\left( {{x^2} - 4x} \right) = P$.
Put this value of P in equation (i),
$
   \Rightarrow P\left( {P - 1} \right) - 20 \\
   \Rightarrow {P^2} - P - 20 \\
$
Now, we will factorise this by using the method of splitting the middle term.
$
   \Rightarrow {P^2} - 5P + 4P - 20 \\
   \Rightarrow P\left( {P - 5} \right) + 4\left( {P - 5} \right) \\
   \Rightarrow \left( {P - 5} \right)\left( {P + 4} \right) \\
$
Here, we have factorised this in the factors ( P – 5 ) and ( P + 4 ).
Now, we will replace the value of P.
$ \Rightarrow \left( {{x^2} - 4x - 5} \right)\left( {{x^2} - 4x + 4} \right)$
Now again, we will do the factors.
\[
   \Rightarrow \left( {{x^2} - 5x + x - 5} \right)\left( {{x^2} - 2x - 2x + 4} \right) \\
   \Rightarrow \left( {x\left( {x - 5} \right) + 1\left( {x - 5} \right)} \right)\left( {x\left( {x - 2} \right) - 2\left( {x - 2} \right)} \right) \\
   \Rightarrow \left( {x - 5} \right)\left( {x + 1} \right)\left( {x - 2} \right)\left( {x - 2} \right) \\
\]
Hence, the factors of $\left( {{x^2} - 4x} \right)\left( {{x^2} - 4x - 1} \right) - 20$ are \[\left( {x - 5} \right)\left( {x + 1} \right){\left( {x - 2} \right)^2}\]

Note:Whenever we ask this type of question, we have to remember some basic points of factorisation of polynomials. First, we have to substitute the value that is common in the given polynomial with a variable. Then we will factorise that polynomial and by getting the factors, we will replace that variable with its original value. After that we will again factorise that polynomial and the factors then formed will be the correct answer of the question.