Question

Solve the following equation:
\[{{x}^{3}}-{{\left( x+1 \right)}^{2}}=2001\]

Answer 1

Hint: In this question, we first need to expand the terms in the given equation. Then rearrange the terms accordingly to simplify it further. Now, factorise the equation so obtained to get the result.

Complete step-by-step answer:
Polynomial: An expression of the form \[{{a}_{0}}{{x}^{n}}+{{a}_{1}}{{x}^{n-1}}+{{a}_{2}}{{x}^{n-2}}+....+{{a}_{n-1}}x+{{a}_{n}}\], where $a_0$, $a_1$,....., $a_n$ are real numbers and n is a non-negative integer, is called a polynomial.
Cubic Polynomial: A polynomial of degree three is called cubic polynomial.
Now, from the given equation in the question we have
\[\Rightarrow {{x}^{3}}-{{\left( x+1 \right)}^{2}}=2001\]
Now, let us expand the square term in the above equation
As we already know the formula that
\[{{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\]
Now, by using the formula the above equation can be further written as
\[\Rightarrow {{x}^{3}}-{{\left( x+1 \right)}^{2}}=2001\]
Now, on substituting the respective values in the formula and expanding it we get,
\[\Rightarrow {{x}^{3}}-\left( {{x}^{2}}+2x+1 \right)=2001\]
Now, this can also be further written as
\[\Rightarrow {{x}^{3}}-{{x}^{2}}-2x-1=2001\]
Let us now rearrange the terms on both the sides
\[\Rightarrow {{x}^{3}}-{{x}^{2}}-2x=2001+1\]
Now, this can be further written in the simplified form as
\[\Rightarrow {{x}^{3}}-{{x}^{2}}-2x=2002\]
Now, let us take the x common on the left hand side
\[\Rightarrow x\left( {{x}^{2}}-x-2 \right)=2002\]
Now, the term inside the bracket can be further written as
\[\Rightarrow x\left( {{x}^{2}}-2x+x-2 \right)=2002\]
Now, by taking the common terms out in the terms inside the bracket we get,
\[\Rightarrow x\left( x\left( x-2 \right)+1\left( x-2 \right) \right)=2002\]
Now, by taking the common terms out and writing it further in the simplified form we get,
\[\Rightarrow x\times \left( x+1 \right)\times \left( x-2 \right)=2002\]
Let us now factorise on the right hand side
\[\Rightarrow x\times \left( x+1 \right)\times \left( x-2 \right)=13\times 14\times 11\]
Now, on comparing the both sides we get,
\[\therefore x=13\]
Hence, the value of x that satisfies the given equation is 13.

Note: Instead of using the factorisation method we can also solve it by using the direct formula to get the roots and then simplify it further to get the result. Both the methods give the same result.
It is important to note that while expanding and rearranging the terms we should not neglect any of the terms because it changes the result completely.