Verified Answer :

  • Subtract the polynomials by distributing the negative sign to each term of the second expression: \(3x^2 + 9x + 13 - x^2 + 2x - 12\).
  • Group the terms by their power of \(x\): \((3x^2 - x^2) + (9x + 2x) + (13 - 12)\).
  • Perform the arithmetic for each coefficient to simplify: \(\boxed{2x^2 + 11x + 1}\).

Explanation

  1. Problem Analysis

The problem asks us to compute the difference between two quadratic expressions: \((3x^2 + 9x + 13)\) and \((x^2 - 2x + 12)\). To subtract these polynomials, we need to subtract the second polynomial from the first one. This is equivalent to adding the additive inverse of the second polynomial to the first one. Specifically, we will distribute the negative sign across every term in the second polynomial and then combine the resulting like terms.

  1. Simplification and Combining Like Terms

To subtract the second polynomial from the first, we rewrite the subtraction as an addition of the negated polynomial:
\[(3x^2 + 9x + 13) - (x^2 - 2x + 12) = 3x^2 + 9x + 13 - x^2 - (-2x) - 12\]
Distributing the negative sign gives us:
\[3x^2 + 9x + 13 - x^2 + 2x - 12\]
Now, we group the terms by their power of \(x\):
\[(3x^2 - x^2) + (9x + 2x) + (13 - 12)\]
Calculating the coefficients:

  • For \(x^2\): \(3 - 1 = 2\), so we get \(2x^2\).
  • For \(x\): \(9 + 2 = 11\), so we get \(11x\).
  • For the constant term: \(13 - 12 = 1\).

Adding these results together, the final expression is \(2x^2 + 11x + 1\).

Examples

In economics, you might use polynomial subtraction to calculate profit functions. If the revenue generated from selling \(x\) units is represented by the quadratic function \(R(x) = 3x^2 + 9x + 13\) and the cost incurred to produce those units is \(C(x) = x^2 - 2x + 12\), then the profit function is the difference between revenue and cost: \(P(x) = R(x) - C(x)\). Solving this gives us \(P(x) = 2x^2 + 11x + 1\), allowing you to quickly analyze profit trends as the number of units produced increases.

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