Simplify $(4 x^3 y-5 x^2 y+6 y)-(9 x^3 y-3 x^2 y-6 y)$
Answer (1)
Distribute the negative sign: ( 4 x 3 y − 5 x 2 y + 6 y ) − ( 9 x 3 y − 3 x 2 y − 6 y ) = 4 x 3 y − 5 x 2 y + 6 y − 9 x 3 y + 3 x 2 y + 6 y .
Combine like terms: 4 x 3 y − 9 x 3 y − 5 x 2 y + 3 x 2 y + 6 y + 6 y = ( 4 − 9 ) x 3 y + ( − 5 + 3 ) x 2 y + ( 6 + 6 ) y .
Simplify the coefficients: ( 4 − 9 ) x 3 y + ( − 5 + 3 ) x 2 y + ( 6 + 6 ) y = − 5 x 3 y − 2 x 2 y + 12 y .
The simplified expression is − 5 x 3 y − 2 x 2 y + 12 y .
Explanation
Understanding the Problem We are given the expression ( 4 x 3 y − 5 x 2 y + 6 y ) − ( 9 x 3 y − 3 x 2 y − 6 y ) and asked to simplify it. This involves combining like terms after distributing the negative sign.
Distributing the Negative Sign First, distribute the negative sign in the second parenthesis: ( 4 x 3 y − 5 x 2 y + 6 y ) − ( 9 x 3 y − 3 x 2 y − 6 y ) = 4 x 3 y − 5 x 2 y + 6 y − 9 x 3 y + 3 x 2 y + 6 y .
Combining Like Terms Next, combine like terms. We group the terms with the same variables together: 4 x 3 y − 9 x 3 y − 5 x 2 y + 3 x 2 y + 6 y + 6 y = ( 4 − 9 ) x 3 y + ( − 5 + 3 ) x 2 y + ( 6 + 6 ) y .
Simplifying Coefficients Now, simplify the coefficients: ( 4 − 9 ) x 3 y + ( − 5 + 3 ) x 2 y + ( 6 + 6 ) y = − 5 x 3 y − 2 x 2 y + 12 y .
Final Answer Therefore, the simplified expression is − 5 x 3 y − 2 x 2 y + 12 y .
Examples
Imagine you're organizing inventory in a warehouse. You start with one set of items and then remove another set. Simplifying polynomial expressions is like combining similar items to see what you have left. For example, if you have 4 x 3 y of one item and remove 9 x 3 y of the same item, you end up with − 5 x 3 y . This helps in managing resources and understanding the net change in inventory.
