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# If a + b + c = 5 and ab + bc + ca = 10, then prove that ${a^3} + {b^3} + {c^3} - 3abc = - 25$

Last updated date: 08th Aug 2024
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Hint: In this particular question use the concept of standard identity such as ${\left( {a + b + c} \right)^2} = {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ca$ and also use the standard identity $\left( {{a^3} + {b^3} + {c^3} - 3abc} \right) = \left( {a + b + c} \right)\left( {{a^2} + {b^2} + {c^2} - \left( {ab + bc + ca} \right)} \right)$ so use these concepts to reach the solution of the question.

Complete step-by-step solution:
Given data:
$a + b + c = 5$...................... (1)
And $ab + bc + ca = 10$................ (2)
Now we have to find out the value of ${a^3} + {b^3} + {c^3} - 3abc = - 25$
Proof –
Consider the LHS of the above equation we have,
$\Rightarrow {a^3} + {b^3} + {c^3} - 3abc$
Now as we all know the common known fact or the standard identity that ${\left( {a + b + c} \right)^2} = {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ca$.
Now the above equation is also written as,
$\Rightarrow {\left( {a + b + c} \right)^2} = {a^2} + {b^2} + {c^2} + 2\left( {ab + bc + ca} \right)$
Now substitute the values from equations (1) and (2) in the above equation we have,
$\Rightarrow {\left( 5 \right)^2} = {a^2} + {b^2} + {c^2} + 2\left( {10} \right)$
Now simplify this we have,
$\Rightarrow 25 - 20 = {a^2} + {b^2} + {c^2}$
$\Rightarrow {a^2} + {b^2} + {c^2} = 5$.................... (3)
Now as we all know a common fact that
$\left( {{a^3} + {b^3} + {c^3} - 3abc} \right) = \left( {a + b + c} \right)\left( {{a^2} + {b^2} + {c^2} - \left( {ab + bc + ca} \right)} \right)$
Now substitute the values from equations (1), (2) and (3) in the above equation we have,
$\Rightarrow \left( {{a^3} + {b^3} + {c^3} - 3abc} \right) = \left( 5 \right)\left( {5 - \left( {10} \right)} \right)$
Now simplify this we have,
$\Rightarrow \left( {{a^3} + {b^3} + {c^3} - 3abc} \right) = \left( 5 \right)\left( { - 5} \right) = - 25$ = RHS
Hence Proved.

Note: Whenever we face such types of questions the key concept we have to remember is that always recall the standard identities which are all stated above which is very useful to solve these types of the problem so from the first identity calculate the value of ${a^2} + {b^2} + {c^2}$, then from the second identity calculate the value of asked expression as above we will get the required answer.