Example What must be added to x² - x + 1 to obtain x³ + 1 ?

Solution: From the wording of the problem we understand that the sum of two expressions here is x³ + 1 of which x² - x + 1 is One. The other can, therefore, be obtained by taking from x³ + 1 the expression x² - x + 1 .

Thus we have :

Example A person possesses $ 7 ab + 5 bc + 4 ac. He spent at different times $ab + bc , $ ½ ac - ½ bc , $ ab + 2 bc - 3/2 ac and $ 3ab - ½ bc + 2 ac. How many dollars does he possess now ?

Solution: In total, he spent

( ab + bc ) + ( ½ ac - ½ bc ) + ( ab + 2 bc - 3/2 ac ) + ( 3 ab - ½ bc + 2 ac )

= ( ab + ab + 3 ab ) + ( bc + 2 bc - bc ) + ( ½ ac - 3/2 ac + 2 ac )

= 5 ab + 2 bc + ac dollars

The person now possesses

= ( 7 ab + 5 bc + 4 ac ) - ( 5 ab + 2 bc + ac )

= 2 ab + 3 bc + 3 ac dollars.