📦 What is a Polynomial?
- A polynomial = math expression with terms like: 3x² + 5x − 7
- TERM = number × variable power | COEFFICIENT = number in front | LIKE TERMS = same variable AND same power
- 3x² and 7x² are LIKE TERMS — BUT — 3x² and 3x are NOT (different powers!)
➕ ADD Polynomials
- 1. Line up LIKE TERMS
- 2. Add the coefficients
- 3. Keep the variable!
- (3x+2) + (5x+4)
- = (3+5)x + (2+4)
- = 8x + 6
➖ SUBTRACT Polynomials
- 1. Distribute the MINUS
- 2. Flip ALL signs after −
- 3. Then add like terms
- (5x+4) − (2x+1)
- = 5x+4−2x−1
- = 3x + 3
✖️ MULTIPLY: FOIL
- F = First: a·c
- O = Outer: a·d
- I = Inner: b·c
- L = Last: b·d
- (x+2)(x+3)
- = x²+5x+6
🔍 FACTOR: GCF
- 1. Find GCF of ALL terms
- 2. Divide each by GCF
- 3. Write: GCF(remaining)
- 6x² + 9x
- GCF = 3x
- = 3x(2x + 3)
FOIL Step-by-Step: (x + 2)(x + 3)
- First: x·x = x² Outer: x·3 = 3x Inner: 2·x = 2x Last: 2·3 = 6
- Combine like terms: x² + 3x + 2x + 6 = x² + 5x + 6
- CHECK: type both forms in Desmos — same graph = same expression!
🔢 Desmos Strategies
3 Ways to Get the Answer!
Strategy 1: Evaluate / Find a Value
- Type with CAPITAL letters to CHECK
- Type (X+2)*(X+3)
- Add slider for X
- Set X=5 → answer = 56
- Also type X^2+5*X+6 at X=5
- Same answer = FOIL is correct!
Strategy 2: Graph / Visualize
- Type y = (x+2)*(x+3)
- Type y = x^2+5*x+6
- SAME GRAPH = they are equal!
- Two expressions = same curve
- = your algebra is correct!
- Compare shapes to verify
Strategy 3: Generate Table
- Type y = x^2+5*x+6 → Table
- Also type factored form → Table
- SAME y-values for each x?
- YES = your factoring is right!
- Use Table to verify GCF too!
- Table is your proof!
Remember This!
- SUBTRACTING: change ALL signs after the minus! −(3x − 2) becomes −3x + 2
- FOIL only works for two binomials (two sets of two terms). Always combine like terms at the end!
- GCF FACTORING CHECK: distribute back — you should get the original expression!