Verified Answer :

  • Distribute the negative sign across the second polynomial: \(-( -8x^2 + 4x + 3) = 8x^2 - 4x - 3\).
  • Combine the results with the first polynomial: \(4x^3 - 12x^2 - 5 + 8x^2 - 4x - 3\).
  • Group and combine like terms by power of \(x\): \(4x^3 + (-12x^2 + 8x^2) - 4x + (-5 - 3)\).
  • Reach the final simplified form: \(\boxed{4x^3 - 4x^2 - 4x - 8}\).

Explanation

  1. Understanding the Problem and Setup

To find the difference between the two polynomials, we need to subtract the second polynomial from the first. We are given:
\[(4x^3 - 12x^2 - 5) - (-8x^2 + 4x + 3)\]
Our task is to perform this subtraction systematically by distributing the negative sign across all terms in the second polynomial.

  1. Distributing the Negative Sign

First, we apply the distributive property of negation to the second polynomial. Each term inside the parentheses after the minus sign changes its sign:
\[- (-8x^2 + 4x + 3) = +8x^2 - 4x - 3\]
Now, we rewrite the entire expression as a single sum by removing the parentheses:
\[4x^3 - 12x^2 - 5 + 8x^2 - 4x - 3\]

  1. Grouping Like Terms

Next, we group the terms with the same power of \(x\) to make the addition easier:

  • The \(x^3\) terms: \(4x^3\)
  • The \(x^2\) terms: \(-12x^2 + 8x^2\)
  • The \(x\) terms: \(-4x\)
  • The constant terms: \(-5 - 3\)

Writing this out:
\[4x^3 + (-12x^2 + 8x^2) - 4x + (-5 - 3)\]

  1. Simplifying and Final Result

Finally, we perform the arithmetic for each group:

  • \(4x^3\) remains as is.
  • For the \(x^2\) terms: \(-12 + 8 = -4\), so we get \(-4x^2\).
  • The \(x\) term remains \(-4x\).
  • For the constants: \(-5 - 3 = -8\).

Combining all these, we obtain the final simplified polynomial:
\[4x^3 - 4x^2 - 4x - 8\]
Thus, the difference is \(\boxed{4x^3 - 4x^2 - 4x - 8}\).

Examples

Polynomial operations are fundamental in various professional fields, such as engineering and finance. For instance, if an inventory manager models a company's total product outflow with one polynomial and its replenishment with another, the difference between these two expressions, calculated using polynomial subtraction, gives the net change in inventory over time. This helps in predicting stock levels, identifying when to reorder, and maintaining operational efficiency.

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