Question

If ${x^2} + \dfrac{1}{{{x^2}}} = 98$, find the value of ${x^3} + \dfrac{1}{{{x^3}}}$.

Answer 1

Hint:In the given question, the value of one polynomial is given and we have to find the value of another polynomial. We will be making use of mathematical identities and formulae and well find the value for the polynomial equation \[{x^3} + \dfrac{1}{{{x^3}}}\].

Complete step by step solution:
\[{x^2} + \dfrac{1}{{{x^2}}} = 98\,\,\,\,\, \cdots \cdots \cdots \left( 1 \right)\]
Adding 2 on both the sides of the above polynomial equation, we get,
\[{x^2} + \dfrac{1}{{{x^2}}} + 2 = 98 + 2\]
Using the identity, \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]on the above polynomial equation,

we get,
\[ \Rightarrow {\left( {x + \dfrac{1}{x}} \right)^2} = 100\,\,\,\, \cdots \cdots \cdots \left( 2 \right)\]
Taking square root on both sides of the above polynomial equation, we get,
\[
\Rightarrow x + \dfrac{1}{x} = \sqrt {100} \\
\Rightarrow x + \dfrac{1}{x} = 10 \\
\]
Now, to find the value of\[{x^3} + \dfrac{1}{{{x^3}}}\], we will use the mathematical identity \[{a^3} +
{b^3} = \left( {a + b} \right)\left( {{a^2} + {b^2} - ab} \right)\], we get,
\[
{x^3} + \dfrac{1}{{{x^3}}} = \left( {x + \dfrac{1}{x}} \right)\left( {{x^2} + \dfrac{1}{{{x^2}}} - x \times
\dfrac{1}{x}} \right) \\
\Rightarrow {x^3} + \dfrac{1}{{{x^3}}} = \left( {x + \dfrac{1}{x}} \right)\left( {{x^2} +
\dfrac{1}{{{x^2}}} - 1} \right) \\
\]
From equations (1) and (2), we will add the values in the above polynomial equation,we get,
\[
{x^3} + \dfrac{1}{{{x^3}}} = \left( {10} \right)\left( {98 - 1} \right) \\
\Rightarrow {x^3} + \dfrac{1}{{{x^3}}} = 10 \times 97 \\
\Rightarrow {x^3} + \dfrac{1}{{{x^3}}} = 970 \\
\]

Formulas used:The formulas used for solving this question are those given below
${a^2} + {b^2} + 2ab = {\left( {a + b} \right)^2}$
${\left( {a + b} \right)^3} = a{}^3 + {b^3} + 3{a^2}b + 3a{b^2}$


Additional information:Monomial is an expression with one term. Binomial is an expression with just two terms. Polynomial is an expression with one or more terms. The numerical term in a factor is the coefficient. Like terms are those terms in which algebraic factors are the same. Unlike terms are those terms in which algebraic factors are different. In case of like terms their sum or difference will also become a like term.

Note:While solving this question remember the expansion of ${\left( {a + b} \right)^2}$ and ${\left( {a
+ b} \right)^3}$. Always try to start solving by expanding the given equation in such questions. Students should be careful when using the sign convention while applying the identities. The raised power of the polynomial expression should be written properly. All the mathematical operations in the solution should be done very carefully by the students.