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In algebra, **like terms** are terms that contain the same variables raised to the same powers. This means they match exactly in their variable parts, not just in the type of variables they have or the order of multiplication. Identifying like terms is crucial in simplifying algebraic expressions because it tells us which terms we can combine.

In our example, the expression is:

• \( 5ab \)
• \( -b^2 \)
• \( 7ab \)
• \( 6b^2 \)

If we look closely, the like terms are:

• \( 5ab \) and \( 7ab \)
• \( -b^2 \) and \( 6b^2 \).

Recognizing these allows us to perform the next step of combining them. Always check the whole part of the term before concluding that they are like terms.

Once we've identified like terms, **combining them** is the next step, which involves adding or subtracting their coefficients while keeping the variable part unchanged. This process streamlines the expression, making it easier to handle in future calculations.

For instance, in our expression:- The terms \( 5ab \) and \( 7ab \) can be combined because they're like terms. Add their coefficients: \[ 5 + 7 = 12 \] So, \( 5ab + 7ab = 12ab \).- Next, for the terms \( -b^2 \) and \( 6b^2 \), combine like this: \[ -1 + 6 = 5 \] Therefore, \(-b^2 + 6b^2 = 5b^2\).

Both operations preserve the variables' integrity, only affecting their coefficients, leading us smoothly to the final expression.

**Polynomial simplification** is the process of reducing a polynomial to its simplest form by combining like terms. This is essential for making the expression easier to evaluate, compare, or solve. A simplified polynomial expression is often more intuitive and manageable.

In the given example, we started with a polynomial expression:\[ 5ab - b^2 + 7ab + 6b^2 \]By identifying and combining like terms, we've reduced it to:\[ 12ab + 5b^2 \]This simpler expression is equivalent to the original. Simplification involves no change to the values involved, only how they are presented. It is fundamental in calculations involving algebraic expressions, either for solving equations or evaluating them in specific scenarios. Thus, mastering simplification skills aids in both understanding and solving algebraic problems more efficiently.