Question

If \[a=-1,b=-2\text{ and }c=-3\]. Find the value of \[ab\left( a-b \right)+bc\left( b-c \right)+ca\left( c-a \right)\].

Answer 1

Hint: In this type of question we have to use the given values of a, b and c to find the value of the given expression. Here we can see that the values of a, b and c are negative so when we simplify the given expression at that time we have to take care about the signs. We know that when we perform multiplication of two negative integers’ results into positive integers, multiplication of a positive and negative integer results in a negative integer. Also when we perform subtraction of two integers it results in subtraction with the sign of the highest number.

Complete step by step solution:
Now, we have to find the value of the algebraic expression \[ab\left( a-b \right)+bc\left( b-c \right)+ca\left( c-a \right)\] and the given values are \[a=-1,b=-2\text{ and }c=-3\].
Let us consider,
\[\Rightarrow ab\left( a-b \right)+bc\left( b-c \right)+ca\left( c-a \right)\]
By substituting the values \[a=-1,b=-2\text{ and }c=-3\] we get,
\[\Rightarrow \left( -1 \right)\left( -2 \right)\left[ \left( -1 \right)-\left( -2 \right) \right]+\left( -2 \right)\left( -3 \right)\left[ \left( -2 \right)-\left( -3 \right) \right]+\left( -3 \right)\left( -1 \right)\left[ \left( -3 \right)-\left( -1 \right) \right]\]
Now, as we know that, multiplication of two negative integers results into positive integers, multiplication of a positive and negative integer results into negative integers. Also when we perform subtraction of two integers it results in subtraction with the sign of highest number. Hence, we can write,
\[\begin{align}
  & \Rightarrow 2\left[ \left( -1 \right)+2 \right]+6\left[ \left( -2 \right)+3 \right]+3\left[ \left( -3 \right)+1 \right] \\
 & \Rightarrow 2\left( 1 \right)+6\left( 1 \right)+3\left( -2 \right) \\
 & \Rightarrow 2+6-6 \\
 & \Rightarrow 2 \\
\end{align}\]
Hence, the value of \[ab\left( a-b \right)+bc\left( b-c \right)+ca\left( c-a \right)\] for the values \[a=-1,b=-2\text{ and }c=-3\] is equal to 2.

Note: In this type of question students have to take care during the calculation. Students have to take care in addition, subtraction and multiplication of negative integers related to the sign of integers.