Verified Answer :

  • Distribute the negative sign across the second polynomial: \(-(w^2 + 5w) = -w^2 - 5w\).
  • Group the like terms for \(w^2\), \(w^1\), and constant terms: \((3w^2 - w^2) + (7w - 5w) + 7\).
  • Combine the coefficients to simplify the expression: \(2w^2 + 2w + 7\).
  • The final simplified polynomial is \(\boxed{2w^2 + 2w + 7}\).

Explanation

  1. Analyzing the problem

To find the difference between the two polynomials \((3w^2 + 7w + 7)\) and \((w^2 + 5w)\), we need to subtract the second polynomial from the first. This involves subtracting each corresponding term of the second polynomial from the first. We will use the standard method of grouping like terms (terms with the same power of \(w\)) to simplify the expression.

  1. Simplifying the expression

We are given the expression: \[(3w^2 + 7w + 7) - (w^2 + 5w)\] First, we distribute the negative sign across the second polynomial: \[3w^2 + 7w + 7 - w^2 - 5w\] Next, we rearrange and group the like terms together: \[(3w^2 - w^2) + (7w - 5w) + 7\] Perform the subtraction for each group: \((3-1)w^2 + (7-5)w + 7 = 2w^2 + 2w + 7\). The result is the simplified polynomial.

Examples

In engineering or physics, we often model physical quantities like distance or velocity using polynomials. If you have an initial position defined by \(P(t) = 3t^2 + 7t + 7\) meters and you subtract an displacement defined by \(Q(t) = t^2 + 5t\) meters, the resulting polynomial \(2t^2 + 2t + 7\) represents your new position at time \(t\). This illustrates how polynomial arithmetic helps in tracking changes in dynamic systems.

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