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In polynomial expressions, it is crucial to identify and group *like terms*. But what are like terms? These are terms in a polynomial that have the same variable raised to the same power. For example, in the expression \(3x^2 - 2x + 5x^2\), the terms \(3x^2\) and \(5x^2\) are like terms because they both contain \(x^2\). They can be combined because they have identical variable parts. Identifying like terms is an essential step when simplifying polynomial expressions, as it helps in keeping the expression organized and manageable. When working on polynomial subtraction, it becomes even more important to look for and combine these like terms as we will be subtracting or adding these terms accordingly. This ensures that the resulting expression is in its simplest form.

When given an expression involving subtraction of polynomials, such as \( \left(x^{2}+2x+1\right)-\left(3x^{2}-6x+2\right) \), you need to *distribute the negative sign* across the second polynomial. This is a crucial step since it changes the signs of every term in the second polynomial. This is done by multiplying each term in the polynomial by \(-1\). As a result, the expression \(- (3x^2 - 6x + 2)\) becomes \(-3x^2 + 6x - 2\). Remember, failing to distribute the negative sign correctly can lead to incorrect simplification, as the negative sign affects the entire polynomial that follows it.In practice:

• A negative sign in front of a bracket means you multiply each term inside the bracket by \(-1\).
• This step converts subtraction into addition of negative terms, laying the accurate path for the next step of combining like terms.

After distributing the negative sign, you end up with a new polynomial expression. The next critical step is to *combine expressions* by gathering all the like terms and simplifying.Consider the expression after distribution: \(x^2 + 2x + 1 - 3x^2 + 6x - 2\). You group terms with the same degree:

• Combine the \(x^2\) terms: \(x^2 - 3x^2\), resulting in \(-2x^2\).
• Combine the \(x\) terms: \(2x + 6x\), equaling \(8x\).
• Combine the constant terms: \(1 - 2\), giving \(-1\).

This results in the simplified form \(-2x^2 + 8x - 1\).The combination of expressions leads to a neater polynomial and ultimately reflects the subtraction process in a simplified manner.