Overview
This lecture explains how to combine like terms in algebraic expressions, especially when negative coefficients are involved, and clarifies related sign notation and rules.
Like Terms and Combining Rules
- Like terms have exactly the same variable(s) with the same exponents.
- Only like terms can be combined by adding or subtracting their coefficients.
- For example, 2y + 4y = 6y, but 2y + 4y² cannot be combined further.
- Terms with different variables (e.g., x and xy) or different exponents (e.g., x and x²) cannot be combined.
Working with Negative Coefficients
- Keep the sign in front of each term when rearranging; it is part of the term.
- When combining, subtract or add coefficients as indicated by the signs.
- Example: 2y + 3xy + 4x - 5x² - 7y + xy + x² → rearrange as 2y - 7y + 3xy + xy + 4x - 5x² + x².
- Combine: (2y - 7y = -5y), (3xy + xy = 4xy), (4x remains), (-5x² + x² = -4x²).
- Final answer: -4x² + 4x - 5y + 4xy (order may be by exponent or variable type).
Minus vs. Negative Signs
- Minus and negative signs both use the "-" symbol; it's context dependent.
- A negative sign is at the start of a term or inside parentheses; a minus sign separates terms.
- In distributing, a negative outside parentheses changes the signs of all terms inside.
Operations with Negative Numbers
- Adding a negative number is the same as subtracting its positive value (6 + -4 = 6 - 4 = 2).
- Subtracting a negative is the same as adding the positive (e.g., -5 - (-7) = -5 + 7 = 2).
- Use parentheses to clarify operations with negatives (e.g., -5 + (-7)).
Expressions vs. Equations
- Expressions do not have an equals sign; equations do.
- Only change the sign of a term when moving it across an equals sign in an equation.
Exponents vs. Multiplication
- Exponent notation: x² means x * x.
- Multiplication notation: 2x means x + x.*
Key Terms & Definitions
- Like Terms — Terms with exactly the same variable(s) and exponents.
- Coefficient — The numerical factor multiplying a variable in a term.
- Negative Coefficient — A coefficient less than zero, indicated by a minus sign before the term.
- Expression — A mathematical phrase with no equals sign.
- Equation — A mathematical statement that shows equality, containing an equals sign.
- Exponent — Indicates how many times a number (base) is multiplied by itself (e.g., x²).
- Distributive Property — Multiplying a number across terms inside parentheses.
Action Items / Next Steps
- Practice combining like terms with both positive and negative coefficients.
- Review operations with negative numbers on the number line.
- Complete any assigned problems on combining like terms and distinguishing expressions from equations.