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In mathematics, we often need to simplify expressions by gathering similar parts. That's where like terms come in. Like terms are terms in a polynomial that have the exact same variable parts. They can only differ by their coefficients, which are constant numbers multiplying the variables.
To identify like terms, look for terms with the same combination of variables raised to the same powers. For example, in the polynomial given,

• \( 5a^3b^2 \) and \( 3a^3b^2 \) qualify as like terms because both involve \( a^3b^2 \).
• Similarly, \( 4ab^3 \) and \( -ab^3 \) are considered like terms since they both contain \( ab^3 \).

By identifying like terms, we can start simplifying the polynomial by calculating the sum or difference of their coefficients. This simplification helps us manage complex expressions more easily.

Once you've combined like terms, the next step often involves arranging these terms in a specific order. Arranging terms helps to better understand and interpret expressions or polynomials. In our exercise, we arrange the polynomial terms in ascending order of the power of \( b \).
To arrange in ascending order, follow these tips:

• Look at each term's degree specific to the variable of interest. Here, that variable is \( b \).
• You want to list terms starting from the smallest exponent of \( b \) to the largest.

For example, consider the combined expression \( -2a^2b + 8a^3b^2 + 3ab^3 \). The smallest power of \( b \) is in \( -2a^2b \) and the largest is in \( 3ab^3 \). Hence, it's correctly arranged in ascending order based on \( b \): \[ -2a^2b + 8a^3b^2 + 3ab^3 \].
This type of arrangement enhances clarity, especially when solving equations or performing further operations.

Combining like terms is a crucial step in simplifying any polynomial. Once you've identified like terms based on their matching variable parts, you add or subtract their coefficients.
In the polynomial from the exercise:

• We first combined \( 5a^3b^2 \) with \( 3a^3b^2 \). The coefficients are added: \( 5 + 3 = 8 \) to get \( 8a^3b^2 \).
• Similarly, \( 4ab^3 \) is combined with \( -ab^3 \). Subtraction here due to the minus gives \( 4 - 1 = 3 \), resulting in \( 3ab^3 \).

By combining like terms, expressions become simpler, and the path to solving equations becomes clearer. Essentially, combining like terms reduces the polynomial to its simplest form, making further mathematical manipulation easier.