1. Basic Definitions
- Polynomial: Expression p(x) = a?xn + a??1xn?¹ + ... + a1x + a0
- Degree: Highest power of variable (n)
- Types by degree:
- Linear: Degree 1 (ax + b)
- Quadratic: Degree 2 (ax² + bx + c)
- Cubic: Degree 3 (ax³ + bx² + cx + d)
- Bi-quadratic: Degree 4 (ax4 + bx² + c)
2. Zeroes/Roots of Polynomial
- For p(x), if p(k) = 0, then k is a zero/root
- Linear: 1 zero
- Quadratic: 2 zeros (may be equal)
- Cubic: 3 zeros
- Degree n: Maximum n zeros
3. Relationship: Zeroes & Coefficients
Quadratic Polynomial: p(x) = ax² + bx + c, a ? 0
If a and ß are zeroes:
Sum of zeroes: a + ß = -b/a
Product of zeroes: aß = c/a
Polynomial: x² - (a+ß)x + aß
Cubic Polynomial: p(x) = ax³ + bx² + cx + d
If a, ß, ? are zeroes:
Sum: a + ß + ? = -b/a
Sum pairwise: aß + ß? + ?a = c/a
Product: aß? = -d/a
4. Division Algorithm for Polynomials
If p(x) and g(x) are polynomials, g(x) ? 0:
p(x) = g(x) × q(x) + r(x)
Where: q(x) = quotient, r(x) = remainder
Important: Degree of r(x) < degree of g(x)
5. Remainder Theorem
If p(x) is divided by (x - a), then remainder = p(a)
Special case: If divided by (ax - b), remainder = p(b/a)
6. Factor Theorem
(x - a) is factor of p(x) ? p(a) = 0
7. Important Identities
- (x + y)² = x² + 2xy + y²
- (x - y)² = x² - 2xy + y²
- x² - y² = (x + y)(x - y)
- (x + y)³ = x³ + y³ + 3xy(x + y)
- (x - y)³ = x³ - y³ - 3xy(x - y)
- x³ + y³ = (x + y)(x² - xy + y²)
- x³ - y³ = (x - y)(x² + xy + y²)
8. Methods to Find Zeroes
For Quadratic:
- Factorization
- Quadratic formula: x = [-b ± v(b² - 4ac)]/2a
- Completing square
For Cubic:
- Trial method to find one zero (check ±1, ±2, etc.)
- Factorize using factor theorem
- Solve remaining quadratic
9. Graph Behavior
- Linear: Straight line
- Quadratic: Parabola (U-shaped)
- Opens upward if a > 0
- Opens downward if a < 0
- Cubic: Curve crossing x-axis up to 3 times
- Number of x-intercepts = Number of real zeroes
10. Important Points
- Polynomial of degree n has at most n zeroes
- Graph of polynomial of degree n can cross x-axis at most n times
- Sum/product formulas work only for standard form polynomials
- A zero may be repeated (equal roots)
- Zeroes may be real or complex (but Class 10 focuses on real)