Simplify: [tex]$\left(7 y^2+3 x y-9\right)-\left(2 y^2+3 x y-5\right)$[/tex]
Answer (2)
Distribute the negative sign: 7 y 2 + 3 x y − 9 − 2 y 2 − 3 x y + 5 .
Group like terms: ( 7 y 2 − 2 y 2 ) + ( 3 x y − 3 x y ) + ( − 9 + 5 ) .
Combine like terms: 5 y 2 + 0 x y − 4 .
Simplify: The final answer is 5 y 2 − 4 .
Explanation
Understanding the Problem We are asked to simplify the expression ( 7 y 2 + 3 x y − 9 ) − ( 2 y 2 + 3 x y − 5 ) . This involves subtracting one polynomial from another. The key is to distribute the negative sign correctly and then combine like terms.
Distributing the Negative Sign First, distribute the negative sign to each term in the second polynomial:
7 y 2 + 3 x y − 9 − 2 y 2 − 3 x y + 5
Grouping Like Terms Next, we group the like terms together:
( 7 y 2 − 2 y 2 ) + ( 3 x y − 3 x y ) + ( − 9 + 5 )
Combining Like Terms Now, combine the like terms:
5 y 2 + 0 x y − 4
This simplifies to:
5 y 2 − 4
Final Answer Therefore, the simplified expression is 5 y 2 − 4 .
Examples
Polynomials are used in various fields such as physics, engineering, computer graphics, and economics. For example, in physics, polynomials can be used to describe the trajectory of a projectile. In economics, they can be used to model cost and revenue functions. Simplifying polynomial expressions allows us to analyze these models more efficiently and make predictions about the systems they represent. For instance, if y represents the number of units sold and the simplified expression 5 y 2 − 4 represents the profit, we can easily calculate the profit for different sales volumes.
To simplify the expression ( 7 y 2 + 3 x y − 9 ) − ( 2 y 2 + 3 x y − 5 ) , we first distribute the negative sign and then combine like terms. The simplified result is 5 y 2 − 4 .
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