Kendall says the sum of $\left(6 x^2+4 x y+2 y^2\right)$ and $\left(4 x^2-2 x y-3 y^2\right)$ is $\left(10 x^2+2 x y-5 y^2\right)$. What error did Kendall make?

AHe combined the like terms $2 y^2$ and $-3 y^2$ incorrectly.
BHe combined the like terms $6 x^2$ and $4 x^2$ incorrectly.
CHe subtracted the polynomials instead of adding.
DHe combined the like terms $4 x y$ and $-2 x y$ incorrectly.

Question

04-11-2026
Mathematics

Answered

Kendall says the sum of $\left(6 x^2+4 x y+2 y^2\right)$ and $\left(4 x^2-2 x y-3 y^2\right)$ is $\left(10 x^2+2 x y-5 y^2\right)$. What error did Kendall make?

AHe combined the like terms $2 y^2$ and $-3 y^2$ incorrectly.
BHe combined the like terms $6 x^2$ and $4 x^2$ incorrectly.
CHe subtracted the polynomials instead of adding.
DHe combined the like terms $4 x y$ and $-2 x y$ incorrectly.

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-04T11:51:15-05:00

Calculate the correct sum of the polynomials by grouping like terms: $(6x^2+4x^2) + (4xy-2xy) + (2y^2-3y^2) = 10x^2+2xy-y^2$.
Compare the correct sum with Kendall's result ($10x^2+2xy-5y^2$).
Observe that the error is specifically in the coefficient of the $y^2$ term ($-1$ vs $-5$).
Final answer: $\boxed{\text{He combined the like terms } 2y^2 \text{ and } -3y^2 \text{ incorrectly.}}$

Explanation

Calculate the Correct Sum

To identify Kendall's error, we first calculate the correct sum of the two given polynomials:
\[P_1 = 6x^2 + 4xy + 2y^2\]
\[P_2 = 4x^2 - 2xy - 3y^2\]
To find the sum $P_1 + P_2$, we combine the like terms:

For the $x^2$ terms: $6x^2 + 4x^2 = 10x^2$
For the $xy$ terms: $4xy - 2xy = 2xy$
For the $y^2$ terms: $2y^2 + (-3y^2) = 2y^2 - 3y^2 = -1y^2$

The correct sum is $10x^2 + 2xy - y^2$.

Identify the Error

Kendall's reported result is $10x^2 + 2xy - 5y^2$.
Comparing the correct result $10x^2 + 2xy - y^2$ with Kendall's result $10x^2 + 2xy - 5y^2$, we can observe that:

The coefficients of $x^2$ are the same (10).
The coefficients of $xy$ are the same (2).
The coefficients of $y^2$ are different: the correct coefficient is $-1$, while Kendall's is $-5$.

This confirms that the error occurred during the addition of the $y^2$ terms, specifically $2y^2 + (-3y^2)$. Kendall calculated this sum incorrectly as $-5y^2$.

Conclusion

Based on the analysis, Kendall made an error while combining the like terms $2y^2$ and $-3y^2$. He arrived at a coefficient of $-5$ instead of $-1$. Thus, the correct option is: "He combined the like terms $2y^2$ and $-3y^2$ incorrectly." $\boxed{\text{He combined the like terms } 2y^2 \text{ and } -3y^2 \text{ incorrectly.}}$

Examples

Polynomial addition is a fundamental skill in algebra with many real-world applications. For example, if a store owner tracks inventory for two different product lines, they might represent the total inventory of each item using polynomials where variables represent types of items. Summing these polynomials allows the owner to quickly calculate the total amount of inventory for all items across both lines, ensuring accurate stock management. Similarly, in physics, vectors representing forces acting on an object can be expressed as polynomials; adding these polynomials helps determine the resultant force, which is essential for predicting the motion of the object.