Marcus finds that $\left(3 x^2-2 y^2+5 x\right)+\left(4 x^2+12 y^2-7 x\right)=7 x^2-10 y^2-2 x$. What error did Marcus make?
A. He combined the terms $5 x$ and $-7 x$ incorrectly.
B. He combined the terms $3 x^2$ and $4 x^2$ incorrectly.
C. He combined the terms $-2 y^2$ and $12 y^2$ incorrectly.
D. He subtracted the polynomials instead of adding.
Answer (1)
Combine like terms in the expression ( 3 x 2 − 2 y 2 + 5 x ) + ( 4 x 2 + 12 y 2 − 7 x ) .
The x 2 terms combine to 7 x 2 .
The y 2 terms combine to 10 y 2 .
The x terms combine to − 2 x .
The correct sum is 7 x 2 + 10 y 2 − 2 x . Marcus's error was in combining the y 2 terms incorrectly. The answer is: He combined the terms − 2 y 2 and 12 y 2 incorrectly.
Explanation
Understanding the Problem We are given the expression ( 3 x 2 − 2 y 2 + 5 x ) + ( 4 x 2 + 12 y 2 − 7 x ) and Marcus's result 7 x 2 − 10 y 2 − 2 x . We need to identify the error Marcus made.
Adding the Polynomials Correctly Let's add the polynomials correctly by combining like terms:
First, combine the x 2 terms: 3 x 2 + 4 x 2 = 7 x 2 .
Next, combine the y 2 terms: − 2 y 2 + 12 y 2 = 10 y 2 .
Finally, combine the x terms: 5 x + ( − 7 x ) = 5 x − 7 x = − 2 x .
So, the correct sum is 7 x 2 + 10 y 2 − 2 x .
Comparing with Marcus's Result Now, let's compare the correct sum 7 x 2 + 10 y 2 − 2 x with Marcus's result 7 x 2 − 10 y 2 − 2 x . We can see that the x 2 and x terms are correct, but the y 2 term is incorrect. Marcus has − 10 y 2 instead of 10 y 2 . This means he combined the y 2 terms incorrectly.
Identifying the Error The error Marcus made was in combining the y 2 terms. He should have added − 2 y 2 and 12 y 2 to get 10 y 2 , but he got − 10 y 2 instead. This indicates an error in adding the coefficients of the y 2 terms.
Conclusion Therefore, the error Marcus made was combining the terms − 2 y 2 and 12 y 2 incorrectly.
Examples
Polynomial addition is used in various fields, such as physics and engineering, to model and analyze systems. For example, when analyzing the trajectory of a projectile, we might use polynomials to represent the horizontal and vertical positions as functions of time. Adding these polynomials allows us to determine the overall displacement of the projectile at any given time. Similarly, in circuit analysis, polynomials can represent the impedance of different components, and adding these polynomials helps determine the total impedance of the circuit.
