Answered

Given polynomials $p(x), q(x)$, and $r(x)$, $p(x)+(q(x)+r(x))=(p(x)+q(x))+r(x)$.



Which equivalent expression would you set up to verify the associative property of addition for $(3x+4)+\left(\left(5x^2-1\right)+(2x+6)\right)$?


A$\left(5x^2-1\right)+((3x+4)+(2x+6))$
B$(3x+4)+\left(5x^2-1\right)+(2x+6)$
C$(3x+4)+\left(5x^2-1\right)$
D$\left((3x+4)+\left(5x^2-1\right)\right)+(2x+6)$

Verified Answer :

  • Identify the polynomials as \(p(x) = 3x + 4\), \(q(x) = 5x^2 - 1\), and \(r(x) = 2x + 6\).
  • Recognize the original expression as \(p(x) + (q(x) + r(x))\).
  • Apply the associative property \(p(x) + (q(x) + r(x)) = (p(x) + q(x)) + r(x)\).
  • Conclude that the equivalent form is \(((3x + 4) + (5x^2 - 1)) + (2x + 6)\), which is option d.

Final answer: \(\boxed{d}\)

Explanation

  1. Understanding the Associative Property

We are given the expression \((3x + 4) + ((5x^2 - 1) + (2x + 6))\) and asked to find an equivalent expression that verifies the associative property of addition. Let the given polynomials be:
\[p(x) = 3x + 4\]
\[q(x) = 5x^2 - 1\]
\[r(x) = 2x + 6\]
The original expression is in the form \(p(x) + (q(x) + r(x))\). The associative property of addition for polynomials states that for any polynomials \(p(x), q(x), r(x)\):
\[p(x) + (q(x) + r(x)) = (p(x) + q(x)) + r(x)\]
This property tells us that the way we group the polynomials does not change the result of the sum.

  1. Identifying the Equivalent Expression

To verify the associative property, we need to transform the original expression from the grouping \(p(x) + (q(x) + r(x))\) to the grouping \((p(x) + q(x)) + r(x)\). Substituting our defined polynomials into this new grouping, we get:
\[((3x + 4) + (5x^2 - 1)) + (2x + 6)\]
Comparing this result to the given options:

A\((5x^2 - 1) + ((3x + 4) + (2x + 6))\)
B\((3x + 4) + (5x^2 - 1) + (2x + 6)\)
C\((3x + 4) + (5x^2 - 1)\)
D\(((3x + 4) + (5x^2 - 1)) + (2x + 6)\)

We see that option (d) matches our derived expression perfectly.

Examples

The associative property of addition is useful when you have multiple items to group or add together in sequence. For example, if you are calculating the total cost of groceries and you have prices in three groups: \(\$15\) in snacks, \(\$30\) in produce, and \(\$25\) in pantry items, you can calculate the total as \(\$15 + (\$30 + \$25) = \$15 + \$55 = \$70\) or as \((\$15 + \$30) + \$25 = \$45 + \$25 = \$70\). The associative property ensures that no matter how you decide to bundle your groups, the final total remains the same.

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