Question
If x + y + z = 6 and xy + yz + zx = 10, then the value of x3 + y3 + z3 - 3xyz is?
Answer: Option A
$$\eqalign{
& {\text{Given}} \cr
& x + y + z = 6 \cr
& xy + yz + zx = 10 \cr
& {\text{To find }}{x^3} + {y^3} + {z^3} - 3xyz{\text{ = ?}} \cr
& \Rightarrow {\text{ Using formula,}} \cr} $$
$$ \Rightarrow {\left( {x + y + z} \right)^2} = $$ $${x^2} \,+$$ $$\, {y^2} \,+ $$ $$\, {z^2} \,+ $$ $$\, 2\left( {xy + yz + zx} \right)$$
$$\eqalign{
& \Rightarrow {6^2} = {x^2} + {y^2} + {z^2} + 2 \times 10 \cr
& \Rightarrow 36 = {x^2} + {y^2} + {z^2} + 20 \cr
& \Rightarrow {x^2} + {y^2} + {z^2} = 16 \cr} $$
$$ \Rightarrow {x^2} + {y^2} + {z^2} - 3xyz = $$ $$\left( {x + y + z} \right)$$ $$\left( {{x^2} + {y^2} + {z^2} - xy - yz - zx} \right)$$
$$\eqalign{
& \Rightarrow {x^2} + {y^2} + {z^2} - 3xyz = 6\left[ {16 - 10} \right] \cr
& \Rightarrow {x^2} + {y^2} + {z^2} - 3xyz = 6 \times 6 \cr
& \Rightarrow {x^2} + {y^2} + {z^2} - 3xyz = 36 \cr} $$
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$$\eqalign{
& {\text{Given}} \cr
& x + y + z = 6 \cr
& xy + yz + zx = 10 \cr
& {\text{To find }}{x^3} + {y^3} + {z^3} - 3xyz{\text{ = ?}} \cr
& \Rightarrow {\text{ Using formula,}} \cr} $$
$$ \Rightarrow {\left( {x + y + z} \right)^2} = $$ $${x^2} \,+$$ $$\, {y^2} \,+ $$ $$\, {z^2} \,+ $$ $$\, 2\left( {xy + yz + zx} \right)$$
$$\eqalign{
& \Rightarrow {6^2} = {x^2} + {y^2} + {z^2} + 2 \times 10 \cr
& \Rightarrow 36 = {x^2} + {y^2} + {z^2} + 20 \cr
& \Rightarrow {x^2} + {y^2} + {z^2} = 16 \cr} $$
$$ \Rightarrow {x^2} + {y^2} + {z^2} - 3xyz = $$ $$\left( {x + y + z} \right)$$ $$\left( {{x^2} + {y^2} + {z^2} - xy - yz - zx} \right)$$
$$\eqalign{
& \Rightarrow {x^2} + {y^2} + {z^2} - 3xyz = 6\left[ {16 - 10} \right] \cr
& \Rightarrow {x^2} + {y^2} + {z^2} - 3xyz = 6 \times 6 \cr
& \Rightarrow {x^2} + {y^2} + {z^2} - 3xyz = 36 \cr} $$
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