Verified Answer :

  • Distribute the subtraction sign across the second polynomial: \(-(-5ab) = +5ab\), \(-(4b^2) = -4b^2\), and \(-(3a^2) = -3a^2\).
  • Combine the expression to get: \(5a^2 + 4ab - 3b^2 + 5ab - 4b^2 - 3a^2\).
  • Group and simplify the like terms: \((5a^2 - 3a^2) + (4ab + 5ab) + (-3b^2 - 4b^2)\).
  • State the simplified result: \(\boxed{2a^2 + 9ab - 7b^2}\).

Explanation

  1. Analysis of the Polynomials

We are given two quadratic polynomials in terms of variables \(a\) and \(b\):
Polynomial 1: \(P_1 = 5a^2 + 4ab - 3b^2\)
Polynomial 2: \(P_2 = -5ab + 4b^2 + 3a^2\)
The objective is to compute the difference \(P_1 - P_2\). This requires subtracting each term of \(P_2\) from the corresponding terms of \(P_1\). To do this accurately, we must distribute the negative sign across every term in the second polynomial.

  1. Distributing and Grouping Like Terms

To compute \((5a^2 + 4ab - 3b^2) - (-5ab + 4b^2 + 3a^2)\), we first remove the parentheses while carefully distributing the minus sign to each term inside the second pair of parentheses.
The expression becomes:
\[5a^2 + 4ab - 3b^2 - (-5ab) - (4b^2) - (3a^2)\]
\[= 5a^2 + 4ab - 3b^2 + 5ab - 4b^2 - 3a^2\]
Now, we group the terms that contain the same variables and powers:

  • For \(a^2\): \(5a^2 - 3a^2 = 2a^2\)
  • For \(ab\): \(4ab + 5ab = 9ab\)
  • For \(b^2\): \(-3b^2 - 4b^2 = -7b^2\)
  1. Final Calculation and Result

By combining the simplified groups calculated in the previous step, we obtain the final resulting polynomial:
\[2a^2 + 9ab - 7b^2\]
This represents the difference between the two given expressions. Thus, the final answer is \(\boxed{2a^2 + 9ab - 7b^2}\).

Examples

Algebraic manipulation of polynomials is foundational for many fields, including physics and economics. For example, if you are modeling the profit of two different product lines as quadratic functions, finding the difference between them allows you to compare their performance directly. By expressing the difference as a single polynomial, you can easily identify the net impact of changing the input variables \(a\) (such as production cost) and \(b\) (such as raw material costs) on the total difference in profit.

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