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In polynomial operations, a key step is combining like terms. "Like terms" are terms within an expression that have the same variable raised to the same power. For example, the terms \(-3z^2\), \(2z^2\), and \(-2z^2\) are like terms because they all contain the variable \(z^2\). Similarly, \(-4z\), \(2z\), and \(3z\) are like terms because they all have the variable \(z\) raised to the first power.

Combining like terms is a methodical process. You simply add or subtract the coefficients (the numerical parts of the term) of similar terms:

• For \(z^2\) terms, we have: \(-3z^2 + 2z^2 - 2z^2 = -3z^2\)
• For \(z\) terms, we simply add them as: \(-4z + 2z + 3z = z\)
• For constant terms, without any variable, sum them up: \(7 - 1 - 7 = -1\)

The goal is to aggregate similar terms together to simplify the polynomial, making it easier to handle in future steps. This step builds a foundation for further simplification.

Distributing a negative sign across an expression involves changing the signs of the terms inside any parentheses that are preceded by a negative. This step is crucial as it ensures that all terms are correctly represented in the expression.

When you see a negative sign in front of a parenthesis, you essentially are distributing \(-1\) to each term inside:

• For example, consider the expression \(-(2z^2 - 3z + 7)\). When you distribute the negative sign, it becomes \(-2z^2 + 3z - 7\).

This transformation is vital before combining like terms because it allows you to correctly adjust each term's coefficient. Always double-check this step to avoid errors in sign, which can lead to incorrect results.

Polynomial simplification is the process of condensing a polynomial expression into its simplest form, where no like terms are left to combine and all operations within the polynomial have been completed.

To simplify a polynomial, follow these essential strategies:

• Ensure all like terms have been combined. As explained earlier, combining like terms means adding their coefficients.
• Expand expressions and distribute any negative signs accurately. This includes removing parentheses and ensuring each term has been considered.

The goal is to reach a form where further operations reveal no additional simplification is possible. The final, simplified expression for the given problem is \(-3z^2 + z - 1\). This stage sets a clear and manageable basis for any further operations such as solving equations or other advanced manipulations.