Overview
This lecture covers how to add, subtract, multiply, and divide polynomial expressions using various algebraic methods.
Adding and Subtracting Polynomials
- Combine like terms (same variables with same exponents) to add or subtract polynomials.
- Distribute negative signs before subtracting.
- Example: (4x² + 5x + 7) + (3x² - 8x + 12) = 7x² - 3x + 19.
- Example: (9x² - 7x + 13) - (5x² + 7x + 14) = 4x² + 27.
Multiplying Polynomials
- Use the distributive property (FOIL for binomials) to multiply polynomials.
- Example: (3x + 5)(2x - 3) = 6x² + 1x - 15.
- Squaring a binomial: (2x - 5)² = 4x² - 20x + 25.
- Multiplying a binomial by a trinomial results in six terms before combining like terms.
- Example: (4x - 2)(x² + 3x - 5) = 4x³ + 10x² - 26x + 10.
- Multiplying two trinomials gives nine terms before combining.
- Example: (3x² - 5x + 7)(2x² + 6x - 4) = 6x⁴ + 8x³ - 28x² + 62x - 28.
Dividing Polynomials
- Can factor numerator and denominator and then cancel common terms.
- Example: (x² + 7x + 12) / (x + 3) = x + 4 after factoring.
- Use long division when factoring is not possible.
- Example: (2x² - x + 6) ÷ (x - 2): divide, multiply, and subtract step by step.
- Synthetic division is faster when the denominator is of the form x - c.
- Write coefficients and perform multiply-add steps with c.
- Example: (2x² - 7x + 6) ÷ (x - 2) = 2x - 3.
Key Terms & Definitions
- Polynomial — An expression of one or more algebraic terms, each including a variable raised to a non-negative integer power.
- Binomial — A polynomial with exactly two terms.
- Trinomial — A polynomial with exactly three terms.
- Like terms — Terms that have identical variable parts and exponents.
- FOIL method — Technique for multiplying two binomials: First, Outside, Inside, Last.
- Synthetic division — Shortcut for dividing polynomials when the divisor is linear (x - c).
Action Items / Next Steps
- Practice combining like terms in polynomial addition and subtraction.
- Try multiplying binomials, binomial and trinomial, and two trinomials.
- Practice division of polynomials using factoring, long division, and synthetic division.