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When subtracting polynomials, we often encounter expressions with parentheses, particularly when one polynomial is subtracted from another. To handle this, it's essential to distribute the negative sign across the terms inside the parentheses. This ensures that each term's sign changes correctly.

Consider the expression: \[(-8.9p^{3} - 2.4p) - (2.1p^{3} + 0.8p^{2} - p)\]Here, the negative sign before the second polynomial indicates subtraction. To "distribute" means to multiply each term inside the parentheses by \(-1\).

• For \(2.1p^3\), it becomes \(-2.1p^3\).
• For \(0.8p^2\), it turns into \(-0.8p^2\).
• Finally, \(-p\) becomes \(+p\) (since subtracting a negative is like adding).

After distributing the negative sign, the expression becomes:\[-8.9p^{3} - 2.4p - 2.1p^{3} - 0.8p^{2} + p\]This step lays the groundwork for combining like terms by aligning all terms for easy comparison.

Combining like terms is crucial for simplifying polynomial expressions, as it allows us to merge similar components into a more concise form. Like terms have identical variable parts, meaning both the variable and its exponent must match.

Now, consider the expression after distributing the negative sign:\[-8.9p^{3} - 2.4p - 2.1p^{3} - 0.8p^{2} + p\]Let's break it down:

• Identify and combine the \(p^3\) terms: \(-8.9p^3 - 2.1p^3 = -11p^3\).
• The \(p^2\) term stands alone: \(-0.8p^2\); no other \(p^2\) terms are present to combine with.
• For the \(p\) terms: \(-2.4p + p = -1.4p\), since adding \(+1p\) reduces the coefficient of \(-2.4p\).

Through combining these like terms, we've successfully condensed the polynomial to a simpler form, preparing it for the final step of simplification.

The final act in polynomial arithmetic is the simplification, where we rewrite the polynomial in its most compact form following the distribution and combination steps.

After combining like terms, the expression becomes:\[-11p^3 - 0.8p^2 - 1.4p\]This is the simplified form, where each term is distinct and no further combination is possible. Each component—\(-11p^3\), \(-0.8p^2\), and \(-1.4p\)—is clearly separated based on the variables and their exponents. This expression provides a clean and concise representation of the original problem, making it easier to handle, analyze, or graph where necessary.

Remember the following key points as you simplify:

• Simplification reduces redundancy in expression.
• It ensures clarity by minimizing the number of terms present in the expression.

Polynomial subtraction is now complete, leaving you with a clear and exact result to work with.