Question

Factorize: \[({x^2} - 3x)({x^2} - 3x - 1) - 20\]

Answer 1

Hint: The method of decomposing the algebraic expressions is called factorization (i.e.) breaking the whole thing into multiple parts. A single entity in the expression which has to be operated with arithmetic operations is called terms. The decomposed terms are called factors, when we multiply these again, we will get the original expression i.e., \[Expression = factor \times factor\].
Few things to know:
Try to reduce the given expression by taking out the common factor.
If the reduced expression is containing perfect or different squares then use algebraic identities to solve it.

Complete step by step answer:
The given expression is\[({x^2} - 3x)({x^2} - 3x - 1) - 20\].
Our aim is to decompose this expression into factors, let \[a = {x^2} - 3x\].
Substituting this in the given expression we get,
\[({x^2} - 3x)({x^2} - 3x - 1) - 20\]\[ = a(a - 1) - 20\]
Simplifying the right hand side of the above equation we will get,
\[ = {a^2} - a - 20\]
Now we got a quadratic equation. Let us solve it.
 \[ = {a^2} - 5a + 4a - 20\]
Let us group it by taking out the common term,
\[ = a(a - 5) + 4(a - 5)\]
Now we can see that the term \[(a - 5)\] is common, let us take it out.
\[ = (a - 5)(a + 4)\]
We have attained the required answer that is a product of factors. Now we have to substitute the value of \[a\] to get back to the original terms.
\[ = ({x^2} - 3x - 5)({x^2} - 3x + 4)\]
Thus, we have factored the expression \[({x^2} - 3x)({x^2} - 3x - 1) - 20\] into product of the factors \[({x^2} - 3x - 5)\] and\[({x^2} - 3x + 4)\].

Note: In the above problem we have used the substitution method by substituting \[a\] to the term\[{x^2} - 3x\]. This will help us to make the expression simpler, which will lead us to solve the problem quickly. After factorization we will re-substitute the term \[a\]to \[{x^2} - 3x\]so that the factors will be in the original terms.