Chapter 2: Polynomials
Key Formulas:
• Quadratic: ax²+bx+c, Sum of zeros = -b/a, Product = c/a
• Cubic: ax³+bx²+cx+d, Sum = -b/a, Sum of products = c/a, Product = -d/a
• Division Algorithm: Dividend = Divisor × Quotient + Remainder
Exercise 2.1
Q1. Find the number of zeroes from the given graphs of y = p(x). (i) Graph doesn't meet x-axis (ii) Meets at 1 point (iii) 3 points (iv) 2 points (v) 4 points (vi) 3 points
SolutionNumber of zeroes = number of x-axis intersections:
(i) 0 zeroes (ii) 1 zero (iii) 3 zeroes
(iv) 2 zeroes (v) 4 zeroes (vi) 3 zeroes
Exercise 2.2
Q1. Find zeroes and verify relationships:
(i) x²-2x-8 (ii) 4s²-4s+1 (iii) 6x²-7x-3 (iv) t²-15 (v) 3x²+x-4
Solution
(i) x²-2x-8 = 0
(x-4)(x+2)=0 → x=4, x=-2
Sum = 4+(-2) = 2 = -(-2)/1 ✓
Product = 4×(-2) = -8 = -8/1 ✓
(ii) 4s²-4s+1 = 0
(2s-1)²=0 → s=1/2, 1/2
Sum = 1/2+1/2 = 1 = -(-4)/4 ✓
Product = 1/4 = 1/4 ✓
(iii) 6x²-7x-3 = 0
6x²-9x+2x-3 = 3x(2x-3)+1(2x-3) = (3x+1)(2x-3)=0
x = -1/3, 3/2
Sum = 7/6 = -(-7)/6 ✓
Product = -1/2 = -3/6 ✓
(iv) t²-15 = 0
t = ±√15
Sum = 0 ✓ Product = -15 ✓
(v) 3x²+x-4 = 0
3x²+4x-3x-4 = x(3x+4)-1(3x+4) = (x-1)(3x+4)=0
x = 1, -4/3
Sum = -1/3 ✓ Product = -4/3 ✓
Q2. Find quadratic polynomials given sum and product of zeroes:
(i) 1/4, 1 (ii) √2, 1/3 (iii) 0, √5 (iv) 1, 1 (v) -1/4, 1/4 (vi) 4, 1
SolutionRequired polynomial: x² - (sum)x + product
(i) x²-x/4+1 or 4x²-x+4
(ii) x²-√2x+1/3 or 3x²-3√2x+1
(iii) x²+√5
(iv) x²-x+1
(v) x²+x/4+1/4 or 4x²+x+1
(vi) x²-4x+1
Exercise 2.3
Q1. Divide and find quotient and remainder:
(i) x³-3x²+5x-3 ÷ x²-2
(ii) x⁴-3x²+4x+5 ÷ x²-x+1
Solution(i) x³-3x²+5x-3 = (x-3)(x²-2) + 7x-9
Quotient: x-3, Remainder: 7x-9
(ii) Quotient: x²+x-3, Remainder: x+8
Q2. Check whether first polynomial is a factor:
(i) t²-3 divides 2t⁴+3t³-2t²-9t-12?
(ii) x²+3x+1 divides 3x⁴+5x³-7x²+2x+2?
Solution(i) Quotient=2t²+3t+4, Remainder=0 → Yes, it is a factor
(ii) Quotient=3x²-4x+2, Remainder=0 → Yes, it is a factor
Q3. Find other zeroes of 3x⁴+6x³-2x²-10x-5 if two zeroes are √(5/3) and -√(5/3).
SolutionFactor from given zeroes: x²-5/3, multiply by 3: 3x²-5
Dividing: (3x⁴+6x³-2x²-10x-5) = (3x²-5)(x²+2x+1)
x²+2x+1 = (x+1)² → Other zeroes: x = -1, -1
Q4. Dividing x³-3x²+x+2 by g(x) gives quotient x-2 and remainder -2x+4. Find g(x).
Solutiong(x)(x-2) = x³-3x²+x+2-(-2x+4) = x³-3x²+3x-2
g(x) = (x³-3x²+3x-2)/(x-2) = x²-x+1
Exercise 2.4 (Optional)
Q1. Verify zeroes of 2x³+x²-5x+2 are 1/2, 1, -2 and verify coefficient relationships.
Solutionp(1/2)=2(1/8)+1/4-5/2+2 = 1/4+1/4-5/2+2 = 0 ✓
p(1)=2+1-5+2 = 0 ✓
p(-2)=-16+4+10+2 = 0 ✓
α+β+γ = 1/2+1+(-2) = -1/2 = -b/a ✓
αβ+βγ+γα = 1/2-2-1 = -5/2 = c/a ✓
αβγ = 1/2×1×(-2) = -1 = -d/a ✓
Q3. If zeroes of x³-3x²+x+1 are a-b, a, a+b, find a and b.
SolutionSum: (a-b)+a+(a+b) = 3a = 3 → a=1
Product: a(a²-b²) = -1 → 1-b²=-1 → b²=2
∴ a=1, b=±√2