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Subtract the polynomials:


  1. $\left(4 x^2-3 x+10\right)-\left(2 x^2+5 x-4\right)$
  2. $\left(-2 x^2+7 x-3\right)-\left(3 x^2-8 x-9\right)$
  3. $\left(6 x^2-1

ight)-\left(2 x^2+4 x-5\right)$

Verified Answer :

  • Distribute the negative sign across the second polynomial in each expression.
  • Group the like terms (\(x^2\) terms, \(x\) terms, and constants) for each expression.
  • Combine the coefficients of these like terms to arrive at the simplified standard form.
  • The results are 4: \(\boxed{2x^2 - 8x + 14}\), 5: \(\boxed{-5x^2 + 15x + 6}\), 6: \(\boxed{4x^2 - 4x + 4}\).

Explanation

  1. Understanding Polynomial Subtraction

We are tasked with simplifying three polynomial expressions, each of which involves the subtraction of two trinomials. The standard method for subtracting polynomials is to distribute the subtraction sign across the second trinomial, changing the sign of each term within that expression, and then combining the resulting like terms. Like terms are terms that have the same variable raised to the same power. After distribution, we will group the \(x^2\) terms, the \(x\) terms, and the constant terms separately to arrive at the simplified expression in standard form \(ax^2 + bx + c\).

  1. Calculating the First Expression

For the first expression:
\((4x^2 - 3x + 10) - (2x^2 + 5x - 4)\)
Distribute the negative sign to each term in the second set of parentheses:
\(= 4x^2 - 3x + 10 - 2x^2 - 5x + 4\)
Now, group the like terms:
\(= (4x^2 - 2x^2) + (-3x - 5x) + (10 + 4)\)
Simplify the coefficients:
\(= 2x^2 - 8x + 14\)

  1. Calculating the Second Expression

For the second expression:
\((-2x^2 + 7x - 3) - (3x^2 - 8x - 9)\)
Distribute the negative sign to each term in the second set of parentheses:
\(= -2x^2 + 7x - 3 - 3x^2 + 8x + 9\)
Now, group the like terms:
\(= (-2x^2 - 3x^2) + (7x + 8x) + (-3 + 9)\)
Simplify the coefficients:
\(= -5x^2 + 15x + 6\)

  1. Calculating the Third Expression

For the third expression:
\((6x^2 - 1) - (2x^2 + 4x - 5)\)
Distribute the negative sign to each term in the second set of parentheses:
\(= 6x^2 - 1 - 2x^2 - 4x + 5\)
Now, group the like terms:
\(= (6x^2 - 2x^2) - 4x + (-1 + 5)\)
Simplify the coefficients:
\(= 4x^2 - 4x + 4\)

  1. Conclusion and Final Answers

The simplified results for the three expressions are:

  1. \(\boxed{2x^2 - 8x + 14}\)
  2. \(\boxed{-5x^2 + 15x + 6}\)
  3. \(\boxed{4x^2 - 4x + 4}\)

Examples

Polynomial subtraction is essential in various fields such as physics and engineering, where we often need to calculate the net effect of two competing forces or signals. For example, if two separate signals in a communication system are represented by quadratic functions of time, \(S_1(t)\) and \(S_2(t)\), the difference \(D(t) = S_1(t) - S_2(t)\) tells us the resultant waveform. Mastering polynomial subtraction allows for the effective modeling and analysis of such dynamic systems, helping experts determine the net output in a wide variety of technological and physical applications.

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