Certainly! Let's walk through the steps in detail to solve the given polynomial expression:
Step 1: Write Subtraction of Polynomial as Addition of the Additive Inverse
Given expression:
[tex]\[
(6m^5 + 3 - m^3 - 4m) - (-m^5 + 2m^3 - 4m + 6)
\][/tex]
Rewrite the subtraction as the addition of the additive inverse:
[tex]\[
(6m^5 + 3 - m^3 - 4m) + (m^5 - 2m^3 + 4m - 6)
\][/tex]
Step 2: Rewrite Terms Subtracted as Addition of the Opposite
Breaking down the terms:
[tex]\[
6m^5 + 3 + (-m^3) + (-4m) + m^5 + (-2m^3) + 4m + (-6)
\][/tex]
Step 3: Group Like Terms
Grouping the like terms together:
[tex]\[
[6m^5 + m^5] + [3 + (-6)] + [(-m^3) + (-2m^3)] + [(-4m) + 4m]
\][/tex]
Step 4: Combine Like Terms
Now, we combine the terms within each group:
1. [tex]\(6m^5 + m^5 = 7m^5\)[/tex]
2. [tex]\(3 + (-6) = -3\)[/tex]
3. [tex]\((-m^3) + (-2m^3) = -3m^3\)[/tex]
4. [tex]\((-4m) + 4m = 0\)[/tex]
Step 5: Write the Resulting Polynomial in Standard Form
Putting it all together, we obtain:
[tex]\[
7m^5 - 3m^3 + 0m - 3
\][/tex]
The resulting polynomial expression is:
[tex]\[
7m^5 - 3m^3 + 0m - 3
\][/tex]
By simplifying further, we can drop the term with a zero coefficient:
[tex]\[
7m^5 - 3m^3 - 3
\][/tex]
Therefore, the final answer is:
[tex]\[
7m^5 - 3m^3 - 3
\][/tex]
