source: ncert_exercises
topic: algebra
class_level: class_10
chapter: ch2
exercise: ex2.2
difficulty: intermediate
Class 10 | Ch2: Polynomials | Exercise 2.2
Key Formulas for quadratic polynomial ax²+bx+c with zeroes α and β: Sum of zeroes: α+β = -b/a Product of zeroes: αβ = c/a Quadratic polynomial with given zeroes: x²-(α+β)x+αβ
Q1. Find zeroes of the following quadratic polynomials and verify relationship between zeroes and coefficients: (i) x²-2x-8 (ii) 4s²-4s+1 (iii) 6x²-3-7x (iv) 4u²+8u (v) t²-15 (vi) 3x²-x-4 Answer: (i) x²-2x-8=(x-4)(x+2). Zeroes: 4,-2. Sum=4+(-2)=2=2/1=−(−2)/1 ✓. Product=4×(−2)=−8=−8/1 ✓. ✅ (ii) 4s²-4s+1=(2s-1)². Zeroes: 1/2,1/2. Sum=1=4/4 ✓. Product=1/4=1/4 ✓. ✅ (iii) 6x²-7x-3=(3x+1)(2x-3). Zeroes: -1/3, 3/2. Sum=-1/3+3/2=7/6=7/6 ✓. Product=-1/3×3/2=-1/2=-3/6 ✓. ✅ (iv) 4u²+8u=4u(u+2). Zeroes: 0,-2. Sum=-2=-8/4 ✓. Product=0=0/4 ✓. ✅ (v) t²-15=(t-√15)(t+√15). Zeroes: √15,-√15. Sum=0 ✓. Product=-15 ✓. ✅ (vi) 3x²-x-4=(3x-4)(x+1). Zeroes: 4/3,-1. Sum=4/3-1=1/3=1/3 ✓. Product=-4/3=-4/3 ✓. ✅
Q2. Find a quadratic polynomial whose sum and product of zeroes are given: (i) 1/4, -1 (ii) √2, 1/3 (iii) 0, √5 (iv) 1, 1 (v) -1/4, 1/4 (vi) 4, 1 Answer: Polynomial = k[x²-(sum)x+product]. (i) x²-x/4-1 → multiply by 4: 4x²-x-4. ✅ (ii) x²-√2x+1/3 → 3x²-3√2x+1. ✅ (iii) x²-0x+√5=x²+√5. ✅ (iv) x²-x+1. ✅ (v) x²+x/4+1/4 → 4x²+x+1. ✅ (vi) x²-4x+1. ✅