Answered

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

To verify the associative property of addition, begin by finding the sum of the polynomials:

$\begin{array}{l}

(3 x+4)+\left(\left(5 x^2-1\right)+(2 x+6)\right) \\

=\square x^2+\square x+\square

\end{array}$

Verified Answer :

  • Identify the polynomials: \(p(x)=3x+4\), \(q(x)=5x^2-1\), and \(r(x)=2x+6\).
  • Simplify the sum inside the parenthesis: \(q(x)+r(x) = 5x^2+2x+5\).
  • Add the result to \(p(x)\): \((3x+4) + (5x^2+2x+5) = 5x^2+5x+9\).
  • Final answer: \(\boxed{5x^2+5x+9}\).

Explanation

  1. Problem Analysis

The problem asks us to verify the associative property of polynomial addition by simplifying the expression:
\[ (3x + 4) + ((5x^2 - 1) + (2x + 6)) \]
We identify the three polynomials as \(p(x) = 3x + 4\), \(q(x) = 5x^2 - 1\), and \(r(x) = 2x + 6\). The expression we are evaluating is \(p(x) + (q(x) + r(x))\).

  1. Inner Sum Calculation

We start by calculating the sum within the inner parentheses, which is \(q(x) + r(x)\):
\[ q(x) + r(x) = (5x^2 - 1) + (2x + 6) \]
By grouping like terms, we combine the \(x^2\) term, the \(x\) term, and the constant terms:
\[ 5x^2 + 2x + (-1 + 6) = 5x^2 + 2x + 5 \]
So, the expression becomes:
\[ (3x + 4) + (5x^2 + 2x + 5) \]

  1. Final Sum Calculation

Now, we add \(p(x)\) to the sum obtained in the previous step:
\[ (3x + 4) + (5x^2 + 2x + 5) \]
Again, we group like terms together:
\[ 5x^2 + (3x + 2x) + (4 + 5) \]
Simplifying the coefficients gives:
\[ 5x^2 + 5x + 9 \]
Comparing this to the target form \(\square x^2 + \square x + \square\), we find the coefficients to be 5, 5, and 9. Thus, the final sum is \(\boxed{5x^2 + 5x + 9}\).

Examples

The associative property of addition is fundamental in algebra for simplifying expressions and solving equations. Just as you might bundle grocery items into different bags to make them easier to carry (but the total weight remains the same), grouping polynomials in different ways allows for more efficient calculation. This concept is widely used in computer science for optimizing algorithms that process large amounts of data, where regrouping operations can significantly speed up computation time.

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