Answered

Complete the tasks to subtract the polynomials vertically.

$\left(1.3 t^3+0.4 t^2-24 t\right)-\left(0.6 t^2+8-18 t\right)$



What is the additive inverse of the polynomial being subtracted?

A$0.6 t^2+(-8)+18 t$
B$-0.6 t^2+(-8)+18 t$
C$-0.6 t^2+8-18 t$
D$-0.6 t^2+(-8)+(-18 t)$

Verified Answer :

  • Identify the polynomial to be subtracted: \(0.6t^2 + 8 - 18t\).
  • Calculate the additive inverse by multiplying each term by \(-1\): \(-(0.6t^2) - (8) - (-18t) = -0.6t^2 - 8 + 18t\).
  • Rewrite the result to match the option format: \(-0.6t^2 + (-8) + 18t\).
  • Select the correct option: \(\boxed{b}\).

Explanation

  1. Analysis of the Polynomial and Additive Inverse Concept

We are given two polynomials to subtract: \((1.3t^3+0.4t^2-24t) - (0.6t^2+8-18t)\). The objective is to identify the additive inverse of the polynomial being subtracted, which is \(P(t) = 0.6t^2 + 8 - 18t\). The additive inverse of any polynomial \(P(t)\) is \(-P(t)\), which is found by multiplying each term of the polynomial by \(-1\).

  1. Calculating the Additive Inverse

To find the additive inverse of \(0.6t^2 + 8 - 18t\), we distribute the negative sign to each term within the expression:
\(-(0.6t^2 + 8 - 18t) = -(0.6t^2) - (8) - (-18t)\).
Calculating each part:

  • The negation of \(0.6t^2\) is \(-0.6t^2\).
  • The negation of \(8\) is \(-8\).
  • The negation of \(-18t\) is \(18t\).

Combining these, we get \(-0.6t^2 - 8 + 18t\). Expressing this in the format used in the options, this is equivalent to \(-0.6t^2 + (-8) + 18t\).

  1. Comparison with Given Options

Now, we evaluate the given options to find the correct additive inverse:

A\(0.6 t^2+(-8)+18 t\) (Incorrect: the first term is positive)
B\(-0.6 t^2+(-8)+18 t\) (Correct: matches our calculation)
C\(-0.6 t^2+8-18 t\) (Incorrect: the terms 8 and -18t were not negated correctly)
D\(-0.6 t^2+(-8)+(-18 t)\) (Incorrect: the term -18t was not negated correctly)

Therefore, the correct additive inverse is \(-0.6t^2 + (-8) + 18t\).

  1. Conclusion

Based on the calculation and comparison with the provided options, the additive inverse of the polynomial \((0.6t^2 + 8 - 18t)\) is \(-0.6t^2 + (-8) + 18t\), which corresponds to option b. The final answer is \(\boxed{b}\).

Examples

In algebra, understanding the additive inverse is crucial for performing operations like subtracting polynomials. Just as you negate numbers to find their additive inverse (e.g., the additive inverse of 5 is -5), you negate entire algebraic expressions to perform subtraction correctly. This process is essential for simplifying complex equations in physics, engineering, and economics, where balancing equations and subtracting variable-based components is a standard procedure for solving real-world problems. By mastering this, you ensure accuracy when simplifying expressions like the one here: \((1.3t^3+0.4t^2-24t) + (-0.6t^2-8+18t)\).

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