Question

If \[a + b + c = 9\] and \[ab + bc + ca = 26\] , find the value of \[{a^3} + {b^3} + {c^3} - 3abc\]
A.27
B.29
C.495
D.729

Answer 1

Hint: Here we have to find the value of the given algebraic expression. For that, we will use the different algebraic identities to find the value of the given algebraic expression. We will substitute all the values in the required algebraic identity and then we will simplify it. Solving it further we will get the value of the required algebraic expression.

Complete step-by-step answer:
First, we will consider the equation:
\[a + b + c = 9\] …….. \[\left( 1 \right)\]
On squaring the terms of both sides of the equation, we get
\[ \Rightarrow {\left( {a + b + c} \right)^2} = {9^2}\]………. \[\left( 2 \right)\]
We know from the algebraic identities that
\[ \Rightarrow {\left( {a + b + c} \right)^2} = {a^2} + {b^2} + {c^2} + 2\left( {ab + bc + ca} \right)\]
Now, we will substitute the value of \[a + b + c = 9\] and the value of \[ab + bc + ca = 26\] in the above equation. Therefore, we get
\[ \Rightarrow {9^2} = {a^2} + {b^2} + {c^2} + 2 \times 26\]
On further simplification, we get
\[ \Rightarrow 81 = {a^2} + {b^2} + {c^2} + 52\]
Now, we will subtract the number 52 from both sides of the equation. So, we get
\[\begin{array}{l} \Rightarrow 81 - 52 = {a^2} + {b^2} + {c^2} + 52 - 52\\ \Rightarrow 29 = {a^2} + {b^2} + {c^2}\end{array}\]
Rewriting the equation, we get
\[ \Rightarrow {a^2} + {b^2} + {c^2} = 29\] …….. \[\left( 3 \right)\]
We know from the algebraic identities that
\[{a^3} + {b^3} + {c^3} - 3abc = \left( {a + b + c} \right)\left( {{a^2} + {b^2} + {c^2} - \left( {ab + bc + ca} \right)} \right)\]
Now, substituting the value of \[a + b + c\], \[ab + bc + ca\] and the value of \[{a^2} + {b^2} + {c^2}\] in the above equation, we get
\[ \Rightarrow {a^3} + {b^3} + {c^3} - 3abc = 9 \times \left( {29 - 26} \right)\]
Now, we will subtract the numbers present inside the bracket.
\[ \Rightarrow {a^3} + {b^3} + {c^3} - 3abc = 9 \times 3\]
On multiplying these numbers, we get
\[ \Rightarrow {a^3} + {b^3} + {c^3} - 3abc = 27\]
Hence, the correct option is option A.

Note: Here we have used the algebraic identity to find the value of the given algebraic expression. The algebraic equations are defined as the equalities that are valid for all values of variables in them. The algebraic identities are used for the factorization of different types of polynomials. In addition, algebraic identities are also used in the simplification of the algebraic expressions.