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Because every monomial is a term, **adding and subtracting monomials** follows the same procedure as combining like terms. Two monomials can be combined only if they are like terms — that is, they share exactly the same variable(s) raised to exactly the same exponent(s). When monomials are like terms, they are combined by adding or subtracting their coefficients while leaving the variable part unchanged. If the monomials are not like terms, they cannot be combined and the expression is already in its simplest form. For example, $$25y^2 + 15y^2 = 40y^2$$ because both monomials contain $$y^2$$, so the coefficients 25 and 15 are added. However, $$u^2v$$, $$5u^2$$, and $$3v^2$$ cannot be combined because each has a different variable structure.

Adding and Subtracting Monomials

An operations specialist is updating a training manual for a billing department that uses algebraic formulas to track expenses. According to the algebraic rules for adding and subtracting monomials, what are the two specific conditions that the variable parts of two monomials must meet in order for them to be combined?

Requirements for Combining Monomials in Technical Formulas

Imagine you are drafting a 'Standard Operating Procedure' (SOP) for your company's data management team. The team needs to simplify algebraic expressions used in resource allocation formulas, such as combining $12x^2$ and $8x^2$. In your own words, describe the two specific criteria that must be met for two monomials to be considered 'like terms' and thus eligible to be added or subtracted. Then, explain the step-by-step process for performing the calculation, clearly stating what happens to the numerical coefficients and the variable parts (including their exponents).

Technical Documentation: Rules for Combining Resource Terms

To simplify the expression $$8a^2 + 12a + 1 + 3a^2 - 5a + 4$$, group and combine the like terms. First, identify the terms with the same variables and exponents: the $$a^2$$-terms ($$8a^2$$ and $$3a^2$$), the $$a$$-terms ($$12a$$ and $$-5a$$), and the constants ($$1$$ and $$4$$). Adding their coefficients yields $$8 + 3 = 11$$ for the $$a^2$$-terms, $$12 - 5 = 7$$ for the $$a$$-terms, and $$1 + 4 = 5$$ for the constants. The fully simplified expression is $$11a^2 + 7a + 5$$.

Combining Like Terms in $$8a^2 + 12a + 1 + 3a^2 - 5a + 4$$

To subtract the like monomials in the expression $$ 16pq^3 - (-7pq^3) $$, first verify that they are like terms. Both terms contain the identical variable part $$ pq^3 $$ (which means $$ p^1q^3 $$), confirming they are like terms. Next, apply the rule for subtracting a negative number, which is equivalent to adding its opposite: $$ 16 - (-7) = 16 + 7 = 23 $$. Combine the coefficients while leaving the variable portion unchanged: $$ 16pq^3 - (-7pq^3) = 23pq^3 $$.

Subtracting $$ 16pq^3 - (-7pq^3) $$

Simplify: $$9x^2 + 5y^2$$

Match each expression to its simplified result.

Two monomials are like terms only if they have the same variable(s) raised to the same _____.

$$15y^4 + 10y^4 = ?$$ What happens to the variable part $$y^4$$?

To add $$5x^2 + 10x^2$$, you add the coefficients and also add the exponents, giving $$15x^4$$.

Case context:
Like terms have the same variable(s) raised to the same exponent(s). Only like terms can be added or subtracted, by combining their coefficients.

Question:
Simplify $$25y^2 + 15y^2 + 3v^2$$. Explain which terms can combine and why.

Sample answer:
$$25y^2$$ and $$15y^2$$ are like terms because they both have $$y^2$$. Add their coefficients: $$25y^2 + 15y^2 = 40y^2$$. The term $$3v^2$$ has a different variable, $$v$$, so it cannot combine with the $$y^2$$ terms. Final answer: $$40y^2 + 3v^2$$.

Key points:
- Like terms must share the same variable and exponent
- $$25y^2 + 15y^2 = 40y^2$$
- $$3v^2$$ has a different variable, so it cannot combine
- Final simplified expression is $$40y^2 + 3v^2$$

Rubric:
Full credit: correctly identifies $$25y^2$$ and $$15y^2$$ as like terms, combines them to $$40y^2$$, and explains why $$3v^2$$ stays separate. Partial credit: combines the like terms but gives a weak or missing explanation for $$3v^2$$. No credit: incorrect combination or no explanation.

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