Question

What is the formula for \[{\left( {x - y - z} \right)^2}\] ?

Answer 1

Hint: Here the square of the term is asked. Squaring a term is nothing but multiplying the same term with itself. So here we will multiply the given term. Then opening the brackets we will multiply the terms separately. After that, we will simplify the terms and write the formula.

Complete step by step answer:
Given is the formula for, \[{\left( {x - y - z} \right)^2}\]
On squaring we get,
\[ \left( {x - y - z} \right)\left( {x - y - z} \right)\]
On opening the bracket,
\[x\left( {x - y - z} \right) - y\left( {x - y - z} \right) - z\left( {x - y - z} \right)\]
Now multiplying the terms inside the brackets,
\[{x^2} - xy - xz - xy + {y^2} - yz - xz + yz + {z^2}\]
Rearranging the terms,
\[{x^2} + {y^2} + {z^2} - 2xy - 2yz - 2xz\]
Taking 2 common from last three terms,
\[{x^2} + {y^2} + {z^2} - 2\left( {xy - yz - xz} \right)\]
So the formula for,
\[{\left( {x - y - z} \right)^2} = {x^2} + {y^2} + {z^2} - 2\left( {xy - yz - xz} \right)\]

Hence, the formula for \[{\left( {x - y - z} \right)^2}\] is \[{x^2} + {y^2} + {z^2} - 2\left( {xy - yz - xz} \right)\].

Note:There is no direct formula for the above identity. We have to obtain the formula. But note that the signs after the multiplication of the brackets should be taken care.Like the product of minus and plus sign is minus, that of plus and plus is plus and of minus and minus is also plus. Also note that, the formula above can be modified as per the requirements. If asked for cubic the same bracket will be multiplied three times.