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When we talk about like terms in algebra, we refer to terms that have the same variable and the same exponent. For example, in the expression (3 + 6a + 7a^2 + a^3) + (4 + 7a - 8a^2 + 6a^3), the terms 6a and 7a are like terms because they both have the variable 'a' raised to the first power. Likewise, 7a^2 and -8a^2 are like terms because they both have the variable 'a' raised to the second power.Recognizing like terms is crucial because only like terms can be added or subtracted directly. We must be careful not to combine terms with different variables or different exponents. For instance, we cannot combine 7a and 7a^2 because they don’t have the same exponent. In polynomial addition, identifying and grouping like terms is an essential first step. Remember, constants (numbers without variables) are also like terms with each other. In our example, the constants are 3 and 4, which can be added together.

Grouping like terms helps simplify polynomial addition by making it easier to see which terms can be combined. Let's consider our example: (3 + 6a + 7a^2 + a^3) + (4 + 7a - 8a^2 + 6a^3). Once we identify the like terms, we group them together. This step involves organizing the expression so that like terms are side by side.Here’s how our example looks when we group like terms:

• Constants: (3 + 4)
• First degree terms: (6a + 7a)
• Second degree terms: (7a^2 - 8a^2)
• Third degree terms: (a^3 + 6a^3)

By separating and grouping each kind of term, we make the addition process straightforward and avoid errors. Grouping like terms highlights the structure of the polynomial and sets the stage for the actual addition.

The final step in polynomial addition is adding the grouped like terms. Now that we have our groups from the previous step, we simply add each group.For example: After grouping, our expression was (3 + 4) + (6a + 7a) + (7a^2 - 8a^2) + (a^3 + 6a^3).Now, let's add these groups:

• Constants: 3 + 4 = 7
• First degree terms: 6a + 7a = 13a
• Second degree terms: 7a^2 - 8a^2 = -a^2
• Third degree terms: a^3 + 6a^3 = 7a^3

Combining all these results gives us the simplified polynomial: 7 + 13a - a^2 + 7a^3.By adding the like terms, we obtain the final polynomial form from the original expression. This form is simpler and reveals the polynomial's structure clearly. Remember, always keep like terms together to ensure accurate addition.