\begin{definition}[Like Terms] Terms in an algebraic expression that have the same variables raised to the same powers. \end{definition}
\begin{note} Like terms can have different coefficients, but their variable parts must be identical. \end{note}
Identifying Like Terms
To identify like terms in an algebraic expression:
- Look for terms with the same variables and exponents.
- Ignore the coefficients; they can be different.
\begin{example} In the expression \$3x^2 + 5x - 2x^2 + 7\$:
- \$3x^2\$ and \$-2x^2\$ are like terms because they both have \$x^2\$.
- \$5x\$ is a different term because it has \$x\$ (not \$x^2\$).
- \$7\$ is a constant term with no variable. \end{example}
Combining Like Terms
Combining like terms involves adding or subtracting their coefficients while keeping the variable part unchanged.
- Group the like terms together.
- Add or subtract their coefficients.
\begin{example} Simplify the expression \$4x^2 + 3x - 2x^2 + 5x + 7\$:
- Group like terms: \$(4x^2 - 2x^2) + (3x + 5x) + 7\$.
- Combine coefficients:
- \$4x^2 - 2x^2 = 2x^2\$
- \$3x + 5x = 8x\$
- The simplified expression is \$2x^2 + 8x + 7\$. \end{example}
\begin{warning} Be careful with negative signs when combining terms. They affect the coefficients. \end{warning}
\begin{self_review} Simplify the expression \$5y^3 - 3y + 2y^3 + 4y - 6\$. \end{self_review}
\begin{tok} How does the concept of like terms illustrate the structure and rules of algebra? Are these rules discovered or invented? \end{tok}
