| --- |
| source: ncert |
| topic: algebra |
| class_level: class_10 |
| chapter: ch2 |
| difficulty: intermediate |
| --- |
| |
| Class 10 | Ch2: Polynomials |
| ZEROES OF POLYNOMIAL: |
| Zero of polynomial p(x): value of x where p(x)=0. |
| Geometrically: x-coordinates where graph of y=p(x) crosses x-axis. |
| Linear polynomial (degree 1): has exactly 1 zero. |
| Quadratic polynomial (degree 2): has at most 2 zeroes. |
| Cubic polynomial (degree 3): has at most 3 zeroes. |
| RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS: |
| For quadratic p(x)=ax^2+bx+c with zeroes alpha and beta: |
| Sum of zeroes: alpha+beta = -b/a. |
| Product of zeroes: alpha×beta = c/a. |
| For cubic p(x)=ax^3+bx^2+cx+d with zeroes alpha, beta, gamma: |
| alpha+beta+gamma = -b/a. |
| alpha*beta+beta*gamma+gamma*alpha = c/a. |
| alpha*beta*gamma = -d/a. |
| FORMING POLYNOMIAL FROM ZEROES: |
| Quadratic with zeroes alpha, beta: p(x)=k[x^2-(alpha+beta)x+alpha*beta]. |
| DIVISION ALGORITHM FOR POLYNOMIALS: |
| p(x) = g(x)×q(x) + r(x). |
| Degree of remainder < degree of divisor. |
| SOLVED EXAMPLES: |
| Example 1: Find zeroes of p(x)=6x^2-3-7x=6x^2-7x-3. |
| 6x^2-7x-3=0. Split: 6x^2-9x+2x-3=3x(2x-3)+1(2x-3)=(3x+1)(2x-3). |
| Zeroes: x=-1/3 and x=3/2. |
| Verify: sum=-1/3+3/2=(-2+9)/6=7/6=-(-7)/6=b/a. Wait: -b/a=-(-7)/6=7/6. Correct. |
| Product=(-1/3)(3/2)=-1/2=c/a=-3/6=-1/2. Correct. |
| Example 2: Form quadratic with zeroes 2 and -3. |
| Sum=2+(-3)=-1. Product=2×(-3)=-6. p(x)=x^2-(-1)x+(-6)=x^2+x-6. |
| Example 3: Cubic zeroes 2,-1,-3. |
| Sum=2-1-3=-2=-b/a → b/a=2. Sum of products=2(-1)+(-1)(-3)+(-3)(2)=-2+3-6=-5=c/a. |
| Product=2×(-1)×(-3)=6=-d/a → d/a=-6. p(x)=x^3+2x^2-5x-6 (taking a=1). |
| COMMON MISTAKES: |
| Sum of zeroes=-b/a (NEGATIVE b over a, not b over a). |
| Product of zeroes=c/a (positive, not negative). |
| Zeroes of polynomial: solve p(x)=0 (set equal to zero, NOT to y). |
