Question

Find the sum of the product of x(x + y – r), y(x – y + r) and z(x – y – z)

Answer 1

Hint: Find the product of each of the three terms separately. Then add up these three quantities and write the sum.

Complete step-by-step answer:
The three quantities are given in terms of factors. First, we need to find the products of the factors for these three terms separately.

i) $\text{x}\left( \text{x + y }-\text{ r} \right)\text{ = }{{\text{x}}^{2}}\text{ + xy }-\text{ xr}$

ii) $\text{y}\left( \text{x }-\text{ y + r} \right)\ \text{= xy }-\text{ }{{\text{y}}^{2}}\text{ + yr}$

iii) $\text{z}\left( \text{x }-\text{ y }-\text{ z} \right)\text{ = xz }-\text{ yz }-\text{

}{{\text{z}}^{2}}$

Now, that the products of the three quantities are obtained, their sum is –

$\begin{align}

  & \left( {{\text{x}}^{2}}\text{ + xy }-\text{ xr} \right)\text{ + }\left( \text{xy }-\text{ }{{\text{y}}^{2}}\text{ + yr} \right)\text{ + }\left( \text{xz }-\text{ yz }-\text{ }{{\text{z}}^{2}} \right) \\

 & =\text{ }{{\text{x}}^{2}}\text{ }-\text{ }{{\text{y}}^{2}}\text{ }-\text{ }{{\text{z}}^{2}}\text{ + 2xy +xz }-\text{ xr + yr }-\text{ yz} \\

\end{align}$

Thus, the sum of the three quantities given in the question is $\text{ }{{\text{x}}^{2}}\text{ }-\text{ }{{\text{y}}^{2}}\text{ }-\text{ }{{\text{z}}^{2}}\text{ + 2xy +xz }-\text{ xr + yr }-\text{ yz}$.


Note: This solution can be done alternatively without finding the products alternatively. Instead, one can directly add the three quantities and then find the product of the quantities under the summation itself. By both the methods, the final summation of the products comes to be the same answer.