Verified Answer :

  • Group the polynomials by like terms: \(p_1(x) + p_2(x) = (2x^2 + 5x + 7) + (-10x^2 - 7x + 1)\).
  • Combine the coefficients of each power of \(x\): \((2 - 10)x^2 + (5 - 7)x + (7 + 1)\).
  • Simplify the resulting coefficients to reach the final sum: \(-8x^2 - 2x + 8\).
  • Final Answer: \(\boxed{-8x^2 - 2x + 8}\).

Explanation

  1. Analysis of the Polynomial Addition Problem

We are given two polynomials, \(p_1(x) = 2x^2 + 5x + 7\) and \(p_2(x) = -10x^2 - 7x + 1\). The objective is to compute the sum of these two polynomials, \(S(x) = p_1(x) + p_2(x)\). Polynomial addition is performed by combining the coefficients of like terms—that is, terms with the same exponent of \(x\). We have terms with \(x^2\), terms with \(x^1\), and constant terms (terms with \(x^0\)).

  1. Summing the Coefficients

To find the sum \(S(x) = (2x^2 + 5x + 7) + (-10x^2 - 7x + 1)\), we group the coefficients of the corresponding powers of \(x\):
\[S(x) = (2x^2 - 10x^2) + (5x - 7x) + (7 + 1).\]
Now, we perform the addition of the coefficients:

  • For \(x^2\): \(2 - 10 = -8\)
  • For \(x\): \(5 - 7 = -2\)
  • Constant term: \(7 + 1 = 8\)

Combining these, we obtain the resulting polynomial: \(S(x) = -8x^2 - 2x + 8\).

  1. Conclusion and Final Result

The resulting polynomial after adding the two given expressions is \(S(x) = -8x^2 - 2x + 8\). Therefore, the sum is \(\boxed{-8x^2 - 2x + 8}\).

Examples

Polynomial addition is frequently used in fields like physics and engineering when modeling the resultant force or displacement of an object. For instance, if you are modeling the trajectory of a projectile where one polynomial represents the horizontal position over time and another represents the atmospheric drift, you can find the actual position by adding the two polynomial expressions together. This foundational algebraic skill allows for the simplification of complex systems by combining simple, manageable components into a single representative model.

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