Question

which example illustrates the commutative property of addition for polynomials?
(2x² + 5x) = -(-2x² - 5x)
(2x² + 5x) + 0 = (2x² + 5x)
(2x² + 5x) + (4x² - 4x) = 2x² + 5x + 4x² - 4x
(2x² + 5x) + (4x² - 4x) = (4x² - 4x) + (2x² + 5x)

Explanation:

Step1: Recall Commutative Property of Addition

The commutative property of addition states that for any two numbers (or polynomials) \(a\) and \(b\), \(a + b = b + a\). This means that the order of adding two quantities does not change the result.

Step2: Analyze Each Option

• First Option: \((2x^{2}+5x)=-(-2x^{2}-5x)\)

This is showing the property of negatives (distributive property of multiplication by \(- 1\)), not the commutative property of addition.

• Second Option: \((2x^{2}+5x)+0=(2x^{2}+5x)\)

This is the identity property of addition (adding zero does not change the number), not the commutative property.

• Third Option: \((2x^{2}+5x)+(4x^{2}-4x)=2x^{2}+5x + 4x^{2}-4x\)

This is just removing the parentheses (associative property of addition in terms of removing grouping, but not commuting the terms), not the commutative property.

• Fourth Option: \((2x^{2}+5x)+(4x^{2}-4x)=(4x^{2}-4x)+(2x^{2}+5x)\)

Here, we have two polynomials \((2x^{2}+5x)\) and \((4x^{2}-4x)\), and their sum is the same when we swap their order, which matches the commutative property of addition \(a + b=b + a\) (where \(a = 2x^{2}+5x\) and \(b = 4x^{2}-4x\)).

Answer:

\((2x^{2}+5x)+(4x^{2}-4x)=(4x^{2}-4x)+(2x^{2}+5x)\) (the fourth option)