Overview
This lecture explains how to simplify polynomials by identifying and combining like terms, with step-by-step examples and practice exercises.
Review of Polynomials
- Polynomials are expressions made of terms added or subtracted together.
- Each term has a number part (coefficient) and a variable part.
- Coefficients of 1 and terms with ( x^0 ) (which equals 1) are often omitted for simplicity.
- Polynomials may have missing or extra terms of various degrees.
Combining Like Terms
- Like terms have exactly the same variable part, including the same exponents.
- Terms with identical variable parts can be combined by adding their coefficients.
- If terms are not like terms, they cannot be combined (e.g., ( x^2 ) and ( x^3 )).
- Order of variables in products doesnât matter (e.g., ( xy ) and ( yx ) are like terms due to the commutative property).
- Exponents must match exactly for terms to be like terms (e.g., ( x^2y ) and ( xy^2 ) are not like terms).
Practice Identifying Like Terms
- ( 2x ) and ( 3x ) are like terms and combine to ( 5x ).
- ( 4x ) and ( 5y ) are not like terms and cannot be combined.
- ( 2x^2 ) and ( -7x^2 ) are like terms and combine to ( -5x^2 ).
- ( 4x^2 ) and ( 6x^3 ) are not like terms.
- ( -5xy ) and ( 8yx ) are like terms and combine to ( 3xy ).
- ( 5x^2y ) and ( 5y^2x ) are not like terms due to different exponents.
Examples of Simplifying Polynomials
- ( x^2 + 6x - x + 10 ) simplifies to ( x^2 + 5x + 10 ).
- ( 16 - 2x^3 + 4x - 10 ) simplifies to ( -2x^3 + 4x + 6 ).
- ( 3x^2 + 10 - 3x + 5x^2 - 4 + x ) simplifies to ( 8x^2 - 2x + 6 ).
Tips for Simplification
- Re-arrange terms to spot like terms more easily.
- Combine like terms by adding/subtracting coefficients; keep variable parts unchanged.
- Treat each term's sign carefully; positive or negative signs are important.
Key Terms & Definitions
- Polynomial â an expression with terms added or subtracted together, each consisting of coefficients and variable parts.
- Term â a product of a coefficient and variable(s) with optional exponents.
- Like Terms â terms with exactly the same variable part and exponents.
- Coefficient â the numerical part of a term.
- Constant Term â a term without variables (just a number).
Action Items / Next Steps
- Practice simplifying polynomials by combining like terms using your own examples or problems from your textbook.