Example Subtract
Example From 3 x^{2}  y^{2}+
z^{2} take away the sum of 5x^{2}  7 y^{2}
+ 3 z^{2} , x^{2} + 5 y^{2}  z^{2}
,  4 x^{2} + y^{2}  z2
Example What must be added to x^{2}
 x + 1 to obtain x^{3 }+ 1 ?
Solution: From the wording of the problem
we understand that the sum of two expressions here is x^{3
}+ 1 of which x^{2}  x + 1 is One. The other can,
therefore, be obtained by taking from x^{3} + 1 the expression
x^{2}  x + 1 .
Thus we have :
Example What must be subtracted from a^{3}
+ 2a^{2} + a  2 to get a^{3} + a + 1 ?
Solution:
Example A person possesses $ 7 ab + 5 bc + 4 ac. He spent
at different times $ab + bc , $ ˝ ac  ˝ bc , $ ab + 2 bc  3/2
ac and $ 3ab  ˝ bc + 2 ac. How many dollars does he possess now
?
Solution: In total, he spent
( ab + bc ) + ( ˝ ac  ˝ bc ) + ( ab + 2 bc  3/2 ac ) + ( 3 ab  ˝ bc + 2 ac )
= ( ab + ab + 3 ab ) + ( bc + 2 bc  bc ) + ( ˝ 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.
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