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Which expression is equivalent to $6 x^3+3 y^2-5 x^3+2 y^2 ?$
A) $6 x^3 y^2$
B) $x^6+3 y^4$
C) $x^3+5 y^2$
D) $6 x^6 y^4$
Answer (1)
Identify like terms in the expression.
Combine the coefficients of the x 3 terms: 6 x 3 − 5 x 3 = x 3 .
Combine the coefficients of the y 2 terms: 3 y 2 + 2 y 2 = 5 y 2 .
The simplified expression is x 3 + 5 y 2 .
Explanation
Understanding the Problem We are given the expression 6 x 3 + 3 y 2 − 5 x 3 + 2 y 2 and asked to find an equivalent expression from the given options.
Identifying Like Terms To find the equivalent expression, we need to combine the like terms. Like terms are terms that have the same variable raised to the same power. In this expression, the like terms are 6 x 3 and − 5 x 3 , and 3 y 2 and 2 y 2 .
Combining x^3 Terms Combine the x 3 terms: 6 x 3 − 5 x 3 = ( 6 − 5 ) x 3 = 1 x 3 = x 3 .
Combining y^2 Terms Combine the y 2 terms: 3 y 2 + 2 y 2 = ( 3 + 2 ) y 2 = 5 y 2 .
Final Simplified Expression Now, combine the simplified terms: x 3 + 5 y 2 .
Selecting the Correct Option Comparing our simplified expression x 3 + 5 y 2 with the given options, we see that it matches option C.
Examples
Understanding how to simplify expressions by combining like terms is a fundamental skill in algebra. For example, if you're calculating the total area of two rectangular gardens with similar dimensions, you might end up with an expression like 2 x 2 + 3 x + x 2 + x . Combining like terms simplifies this to 3 x 2 + 4 x , making it easier to understand and calculate the total area.
