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In algebra, 'like terms' are terms that have the same variables raised to the same power. This means they can be added or subtracted just like regular numbers. For example, in the expression \(3r^{2} + 5r^{2}\), both terms are like terms because they share the same variable (r) raised to the same power (2). So, you can add their coefficients: \(3+5 = 8\), making the simplified expression \(8r^{2}\).
Like terms simplify the process of polynomial addition because only coefficients need to be added.
This is why it's important to identify them before performing operations. In the exercise, terms like \(12r^{5}\) and \(-8r^{5}\)\ or \(-7r^{3}\) and \(3r^{3}\)\ are considered like terms.

A 'coefficient' is the numerical part of a term that is multiplied by the variable. For example, in the term \(12r^{5}\), the number 12 is the coefficient. It tells you how many of that term you have. Coefficients play a crucial role when you're adding or subtracting polynomials.
When you have like terms, you only add or subtract the coefficients and keep the variable part the same. For instance, in the polynomial \(12r^{5} + 11r^{4} - 7r^{3} - 2r^{2}\), the coefficient of \(r^{5}\) is 12, \(r^{4}\) is 11, \(-7\) for \(r^{3}\), and \(-2\) for \(r^{2}\).
Understanding coefficients allows you to add polynomials accurately, like in the exercise, where the coefficients of like terms are combined to simplify the expression.

Simplifying expressions involves combining like terms to make an expression more concise and easier to work with. The final step in polynomial addition is simplifying the expression. This means adding the coefficients of like terms and writing the result in a simplified form.
In our exercise, we simplified the expression by aligning like terms and then combining their coefficients: \(12r^{5} - 8r^{5}\) yields \(4r^{5}\), \(-7r^{3} + 3r^{3}\) yields \(-4r^{3}\), and because \(-2r^{2} + 2r^{2}\) equals zero, there's no \(r^{2}\) term in the simplified expression.
Therefore, the simplified expression is \(4r^{5} + 11r^{4} - 4r^{3}\). Simplifying expressions makes them easier to understand and use in further mathematical operations.