Answered

Find the difference.

$\left(0.4 k^3-2.5 k\right)-\left(2.4 k^3+3 k^2-1.2 k\right)=$


A$-2 k^3+3 k^2-3.7 k$
B$-2 k^3-3 k^2-1.3 k$
C$-2.8 k^3-3 k^2-3.7 k$
D$-2.8 k^3+3 k^2-1.3 k$

Verified Answer :

  • Distribute the negative sign across the second polynomial: \(-(2.4k^3 + 3k^2 - 1.2k) = -2.4k^3 - 3k^2 + 1.2k\).
  • Combine like terms: \((0.4k^3 - 2.4k^3) - 3k^2 + (-2.5k + 1.2k)\).
  • Perform the arithmetic to obtain the simplified polynomial: \(-2k^3 - 3k^2 - 1.3k\).
  • The correct option is \(\boxed{b}\).

Explanation

  1. Analyzing the expression

To find the difference between the two given polynomials, we perform the subtraction of the second polynomial from the first. The expressions are:
\[(0.4k^3 - 2.5k) - (2.4k^3 + 3k^2 - 1.2k)\]
Our goal is to simplify this by distributing the negative sign and combining like terms. Like terms are those that share the same exponent for the variable \(k\).

  1. Simplifying the expression

First, we distribute the negative sign across every term in the second polynomial. This changes the sign of each term inside the parenthesis:
\[-(2.4k^3 + 3k^2 - 1.2k) = -2.4k^3 - 3k^2 + 1.2k\]
Now, we rewrite the entire expression as a sum of all individual terms:
\[0.4k^3 - 2.5k - 2.4k^3 - 3k^2 + 1.2k\]
Next, we group the like terms together based on their powers of \(k\):
\[(0.4k^3 - 2.4k^3) + (-3k^2) + (-2.5k + 1.2k)\]

  1. Final calculation and matching the result

Now we calculate the coefficients for each power of \(k\):

  • For \(k^3\): \(0.4 - 2.4 = -2.0\)
  • For \(k^2\): \(-3\) (since there was no \(k^2\) term in the first part)
  • For \(k\): \(-2.5 + 1.2 = -1.3\)

Combining these results, we get the final polynomial:
\[-2k^3 - 3k^2 - 1.3k\]
Comparing this result to the given options:

A\(-2k^3 + 3k^2 - 3.7k\)
B\(-2k^3 - 3k^2 - 1.3k\)
C\(-2.8k^3 - 3k^2 - 3.7k\)
D\(-2.8k^3 + 3k^2 - 1.3k\)

The result matches option b.

Examples

Understanding how to simplify polynomial expressions is essential in fields like physics and engineering, where complex systems are often modeled using polynomials. For example, if you are calculating the total displacement of an object affected by multiple forces acting at different times, you might end up with an expression like this. Combining these forces into a single simplified polynomial allows for easier calculation of speed or position at any given time, making complex physical modeling much more manageable and efficient.