a) [tex]$x-2 y-3$[/tex]
b) [tex]$x+2 y$[/tex]
c) [tex]$-x \pm 2 y$[/tex]
d) [tex]$x+2 y-$[/tex]
24. [tex]$4 a b+\left(1-a^2\right)\left(1-b^2\right)=[/tex]
Answer (1)
Expand the product: ( 1 − a 2 ) ( 1 − b 2 ) = 1 − a 2 − b 2 + a 2 b 2 .
Substitute back into the original expression: 4 ab + ( 1 − a 2 − b 2 + a 2 b 2 ) .
Rearrange the terms: a 2 b 2 − a 2 − b 2 + 4 ab + 1 .
Rewrite and factor using the difference of squares: ( ab + a − b + 1 ) ( ab − a + b + 1 ) .
Explanation
Understanding the Problem We are given the expression 4 ab + ( 1 − a 2 ) ( 1 − b 2 ) and asked to simplify it. To do this, we will expand the product and combine like terms.
Expanding the Product First, let's expand the product ( 1 − a 2 ) ( 1 − b 2 ) : ( 1 − a 2 ) ( 1 − b 2 ) = 1 − a 2 − b 2 + a 2 b 2
Substituting and Combining Now, substitute this back into the original expression: 4 ab + ( 1 − a 2 ) ( 1 − b 2 ) = 4 ab + ( 1 − a 2 − b 2 + a 2 b 2 ) = 4 ab + 1 − a 2 − b 2 + a 2 b 2
Simplifying the Expression Rearranging the terms, we get: a 2 b 2 − a 2 − b 2 + 4 ab + 1 We can rewrite this expression as: a 2 b 2 − 2 ab + 1 − a 2 + 2 ab − b 2 ( ab − 1 ) 2 − ( a 2 − 2 ab + b 2 ) ( ab − 1 ) 2 − ( a − b ) 2 Using the difference of squares formula, x 2 − y 2 = ( x + y ) ( x − y ) , we have: [( ab − 1 ) + ( a − b )] [( ab − 1 ) − ( a − b )] [ ab + a − b − 1 ] [ ab − a + b − 1 ] This doesn't seem to simplify further in an obvious way. Let's try a different approach. Notice that: a 2 b 2 − a 2 − b 2 + 1 + 4 ab = ( a 2 + 1 ) ( b 2 + 1 ) − a 2 − b 2 − 1 + 4 ab + 1 − a 2 b 2 = a 2 b 2 + a 2 + b 2 + 1 − a 2 − b 2 − a 2 b 2 + 4 ab = ( a 2 b 2 − a 2 − b 2 + 1 ) + 4 ab ( a 2 − 1 ) ( b 2 − 1 ) + 4 ab = a 2 b 2 − a 2 − b 2 + 1 + 4 ab Consider ( a − b ) 2 = a 2 − 2 ab + b 2 and ( a + b ) 2 = a 2 + 2 ab + b 2 . We can rewrite the expression as: − ( a 2 − 2 ab + b 2 ) + 2 ab + 1 + a 2 b 2 = − ( a − b ) 2 + 2 ab + 1 + a 2 b 2 This also doesn't seem to lead to a simple factorization. Let's try another rearrangement: a 2 b 2 + 4 ab + 4 − a 2 − b 2 − 3 = ( ab + 2 ) 2 − ( a 2 + b 2 + 3 ) Let's go back to the original expression and try completing the square: 4 ab + 1 − a 2 − b 2 + a 2 b 2 = − ( a 2 − 4 ab + b 2 ) + 1 + a 2 b 2 = − ( a − b ) 2 + 2 ab + 1 + a 2 b 2 = − ( a 2 − 4 ab + b 2 ) + 1 − a 2 b 2 = − ( a − b ) 2 + 2 ab + 1 + a 2 b 2 Notice that if we have ( ab + 1 ) 2 = a 2 b 2 + 2 ab + 1 , then 4 ab + 1 − a 2 − b 2 + a 2 b 2 = ( ab + 1 ) 2 − ( a 2 + b 2 − 2 ab ) = ( ab + 1 ) 2 − ( a − b ) 2 Using the difference of squares, we have: [( ab + 1 ) + ( a − b )] [( ab + 1 ) − ( a − b )] = ( ab + a − b + 1 ) ( ab − a + b + 1 ) This is the simplified form of the expression.
Final Answer Therefore, the simplified expression is ( ab + a − b + 1 ) ( ab − a + b + 1 ) .
Examples
This type of algebraic simplification can be useful in physics when dealing with complex equations involving multiple variables. For example, when analyzing the motion of objects under certain forces, simplifying expressions can reveal underlying symmetries or conserved quantities, making the problem easier to solve. It also helps in optimizing algorithms in computer graphics, where efficient calculations are crucial for rendering images quickly.
