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When dealing with algebraic expressions, a critical step is to simplify them to make them easier to understand and work with. Simplifying expressions often involves combining like terms, which means identifying and merging terms that have the same variable and exponent. In our exercise, the terms

• -25x^3
• -7x^3

are like terms because they both contain the variable \(x\) raised to the power of 3.
To simplify, you focus only on the coefficients, which are the numerical parts of these terms. You then perform the mathematical operation indicated, in this case, addition of the coefficients.
Thus, by adding -25 and -7, you arrive at the coefficient of -32 for the variable part \(x^3\), completing the simplification. This process helps in tidying up expressions so that future calculations become more straightforward.

Algebraic operations refer to the mathematical processes performed on algebraic expressions, including addition, subtraction, multiplication, and division. These operations follow similar rules as arithmetic operations but often involve variables. When handling algebraic expressions such as

• -25x^3
• -7x^3

addition happens only between like terms. Like terms are crucial because they allow us to apply these operations correctly.
For instance, in our exercise, the operation requires adding the coefficients of the like terms. Thus, you calculate \[-25 + (-7) = -32\]This operation allows you to combine the two terms into a single, simpler term, which is \(-32x^3\).
Understanding how to execute these operations accurately is fundamental to mastering algebra.

In algebra, coefficients are the numbers placed before a variable in terms like \(-25x^3\). They represent the multiplication factor of the variable part of the term, essentially scaling the variable. In an expression, finding and using the coefficients appropriately is vital when combining like terms.

• In the expression \(-25x^3 - 7x^3\), the coefficients are -25 and -7.
• The variable part \(x^3\) stays constant while you add the coefficients.

Adding the coefficients \(-25 + (-7)\),gives us \(-32\),which is then paired with the unchanged variable part \(x^3\) to yield the simplified expression of \(-32x^3\).
Understanding coefficients helps in visualizing how each part of an algebraic term contributes to the overall value of expressions.