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Jennifer is rewriting a polynomial by combining like terms. Which terms does she still need to combine to finish rewriting the polynomial?
[tex]4 c^2 d-7 c^2 d^2+3 d^2+c^2 d^2-4 c d^2-c^2[/tex]
Answer (2)
Identify like terms: − 7 c 2 d 2 and c 2 d 2 are like terms.
Combine the like terms: − 7 c 2 d 2 + c 2 d 2 = − 6 c 2 d 2 .
Rewrite the polynomial: 4 c 2 d − 6 c 2 d 2 + 3 d 2 − 4 c d 2 − c 2 .
The terms that needed to be combined were − 7 c 2 d 2 and c 2 d 2 .
Explanation
Understanding the Problem We are given the polynomial 4 c 2 d − 7 c 2 d 2 + 3 d 2 + c 2 d 2 − 4 c d 2 − c 2 . Jennifer is combining like terms, and we need to identify which terms can still be combined.
Identifying Like Terms Like terms are terms that have the same variables raised to the same powers. In the given polynomial, we look for terms with the same variable combinations.
Combining Like Terms We observe that the terms − 7 c 2 d 2 and c 2 d 2 are like terms. These terms can be combined. Also, there are no other like terms in the polynomial.
Simplifying the Polynomial Combining the like terms − 7 c 2 d 2 and c 2 d 2 , we have ( − 7 + 1 ) c 2 d 2 = − 6 c 2 d 2 . So, the polynomial becomes 4 c 2 d − 6 c 2 d 2 + 3 d 2 − 4 c d 2 − c 2 .
Final Answer The terms that Jennifer still needs to combine are − 7 c 2 d 2 and c 2 d 2 .
Examples
Combining like terms is a fundamental skill in algebra, used in various real-world applications. For instance, imagine you are calculating the total cost of materials for a construction project. If you have multiple quantities of the same material with different prices, combining like terms allows you to simplify the calculation and determine the overall expense efficiently. Similarly, in physics, when analyzing forces acting on an object, you combine force components in the same direction to find the net force. This process simplifies the analysis and helps predict the object's motion accurately. Combining like terms helps in resource allocation, financial analysis, and engineering calculations.
Jennifer needs to combine the terms − 7 c 2 d 2 and c 2 d 2 which will simplify to − 6 c 2 d 2 . The rewritten polynomial will be 4 c 2 d − 6 c 2 d 2 + 3 d 2 − 4 c d 2 − c 2 .
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