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Which example illustrates the commutative property of addition for polynomials?

Check the final answer first, then review the worked steps.

Problem

Which example illustrates the commutative property of addition for polynomials?

Answer

\((2x^2 + 5x) + (4x^2 - 4x) = (4x^2 - 4x) + (2x^2 + 5x)\)

Step-by-step solution

  1. Understand the Commutative Property of Addition: The commutative property of addition states that the order in which two numbers (or expressions) are added does not change the sum. For any two numbers $a$ and $b$, $a + b = b + a$. This property also applies to polynomials.
  1. Analyze Each Option: We need to examine each given equation to see which one demonstrates the commutative property of addition for polynomials.
  1. Option 1 Analysis: The first option is $(2x^2 + 5x) + 0 = (2x^2 + 5x)$. This equation illustrates the additive identity property, which states that any number (or expression) added to zero equals itself. It does not show that the order of addition can be changed.
  1. Option 2 Analysis: The second option is $(2x^2 + 5x) = -(-2x^2 - 5x)$. This equation demonstrates the property of additive inverses, where a number is equal to the negative of its opposite. It does not involve the commutative property of addition.
  1. Option 3 Analysis: The third option is $(2x^2 + 5x) + (4x^2 - 4x) = 2x^2 + 5x + 4x^2 - 4x$. This equation shows the addition of two polynomials. While the right side is the result of removing the parentheses, it does not show a change in the order of the terms being added. It is an illustration of the associative property or simply the removal of parentheses when adding polynomials.
  1. Option 4 Analysis: The fourth option is $(2x^2 + 5x) + (4x^2 - 4x) = (4x^2 - 4x) + (2x^2 + 5x)$. This equation shows the sum of two polynomials, $(2x^2 + 5x)$ and $(4x^2 - 4x)$. On the left side, $(2x^2 + 5x)$ is added first, and on the right side, $(4x^2 - 4x)$ is added first. The order of the polynomials being added has been reversed, but the sum remains the same. This is a direct application of the commutative property of addition for polynomials.