Which example illustrates the commutative property of addition for polynomials?

A$\left(2 x^2+5 x\right)+\left(4 x^2-4 x\right)=\left(4 x^2-4 x\right)+\left(2 x^2+5 x\right)$
B$\left(2 x^2+5 x\right)+\left(4 x^2-4 x\right)=2 x^2+5 x+4 x^2-4 x$
C$\left(2 x^2+5 x\right)=-\left(-2 x^2-5 x\right)$
D$\left(2 x^2+5 x\right)+0=\left(2 x^2+5 x\right)$

Question

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04-11-2026
Mathematics

Answered

Which example illustrates the commutative property of addition for polynomials?

A$\left(2 x^2+5 x\right)+\left(4 x^2-4 x\right)=\left(4 x^2-4 x\right)+\left(2 x^2+5 x\right)$
B$\left(2 x^2+5 x\right)+\left(4 x^2-4 x\right)=2 x^2+5 x+4 x^2-4 x$
C$\left(2 x^2+5 x\right)=-\left(-2 x^2-5 x\right)$
D$\left(2 x^2+5 x\right)+0=\left(2 x^2+5 x\right)$

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-04T11:48:40-05:00

The commutative property of addition states that for any two polynomials $P$ and $Q$, $P + Q = Q + P$.
Option A shows that $\left(2x^2 + 5x\right) + \left(4x^2 - 4x\right) = \left(4x^2 - 4x\right) + \left(2x^2 + 5x\right)$, which follows this form.
Options B, C, and D represent the removal of parentheses, the distributive property, and the additive identity property, respectively.
The correct choice is A.
Final Answer: $\boxed{A}$

Explanation

Understanding the Commutative Property of Addition

The commutative property of addition for any two expressions, including polynomials, states that changing the order of the addends does not change the sum. Formally, for any polynomials $P$ and $Q$, the property is expressed as:
\[P + Q = Q + P\]
We need to evaluate each of the four given options against this definition.

Evaluating the Options

Let's analyze the provided options:

Option A: $\left(2x^2 + 5x\right) + \left(4x^2 - 4x\right) = \left(4x^2 - 4x\right) + \left(2x^2 + 5x\right)$. This clearly shows two polynomials being added, where the order is swapped, fitting the form $P + Q = Q + P$.
Option B: $\left(2x^2 + 5x\right) + \left(4x^2 - 4x\right) = 2x^2 + 5x + 4x^2 - 4x$. This illustrates the removal of parentheses, not the commutative property.
Option C: $\left(2x^2 + 5x\right) = -\left(-2x^2 - 5x\right)$. This illustrates the property of multiplying by $-1$, or the distributive property.
Option D: $\left(2x^2 + 5x\right) + 0 = \left(2x^2 + 5x\right)$. This illustrates the additive identity property.

Conclusion

Based on our analysis, Option A is the only one that correctly represents the commutative property of addition, as it demonstrates that swapping the position of the two polynomial addends results in an equivalent expression. Thus, option A is the correct answer.

Examples

The commutative property is fundamental in simplifying algebraic expressions. For instance, when you are rearranging terms in a long polynomial to group similar powers of $x$ together, you are implicitly using the commutative property of addition. Being able to change the order of terms makes calculations much more manageable and less prone to errors.