Exercise 2.1 Solutions
1. The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.
The number of zeroes of a polynomial is equal to the number of points where the graph intersects the x-axis.
(i) The graph intersects the x-axis at 0 points.
Number of zeroes = 0
(ii) The graph intersects the x-axis at 1 point.
Number of zeroes = 1
(iii) The graph intersects the x-axis at 3 points.
Number of zeroes = 3
(iv) The graph intersects the x-axis at 2 points.
Number of zeroes = 2
(v) The graph intersects the x-axis at 4 points.
Number of zeroes = 4
(vi) The graph intersects the x-axis at 3 points.
Number of zeroes = 3
Exercise 2.2 Solutions
1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) x² - 2x - 8
x² - 2x - 8 = x² - 4x + 2x - 8
= x(x - 4) + 2(x - 4)
= (x + 2)(x - 4)
Zeroes: x = -2, 4
Verification:
Sum of zeroes = -2 + 4 = 2 = -(-2)/1 = -b/a
Product of zeroes = (-2) × 4 = -8 = -8/1 = c/a
(ii) 4s² - 4s + 1
4s² - 4s + 1 = (2s)² - 2×2s×1 + 1²
= (2s - 1)²
Zeroes: s = 1/2, 1/2 (equal zeroes)
Verification:
Sum of zeroes = 1/2 + 1/2 = 1 = -(-4)/4 = -b/a
Product of zeroes = (1/2) × (1/2) = 1/4 = 1/4 = c/a
(iii) 6x² - 3 - 7x
Rewriting: 6x² - 7x - 3
= 6x² - 9x + 2x - 3
= 3x(2x - 3) + 1(2x - 3)
= (3x + 1)(2x - 3)
Zeroes: x = -1/3, 3/2
Verification:
Sum of zeroes = -1/3 + 3/2 = (-2 + 9)/6 = 7/6 = -(-7)/6 = -b/a
Product of zeroes = (-1/3) × (3/2) = -1/2 = -3/6 = c/a
(iv) 4u² + 8u
4u² + 8u = 4u(u + 2)
Zeroes: u = 0, -2
Verification:
Sum of zeroes = 0 + (-2) = -2 = -8/4 = -b/a
Product of zeroes = 0 × (-2) = 0 = 0/4 = c/a
(v) t² - 15
t² - 15 = (t - √15)(t + √15)
Zeroes: t = √15, -√15
Verification:
Sum of zeroes = √15 + (-√15) = 0 = -0/1 = -b/a
Product of zeroes = √15 × (-√15) = -15 = -15/1 = c/a
(vi) 3x² - x - 4
3x² - x - 4 = 3x² - 4x + 3x - 4
= x(3x - 4) + 1(3x - 4)
= (x + 1)(3x - 4)
Zeroes: x = -1, 4/3
Verification:
Sum of zeroes = -1 + 4/3 = (-3 + 4)/3 = 1/3 = -(-1)/3 = -b/a
Product of zeroes = (-1) × (4/3) = -4/3 = -4/3 = c/a
2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
General form of quadratic polynomial: x² - (sum of zeroes)x + (product of zeroes)
(i) 1/4, -1
Sum of zeroes = 1/4, Product of zeroes = -1
Polynomial: x² - (1/4)x + (-1)
= x² - (1/4)x - 1
Multiplying by 4: 4x² - x - 4
(ii) √2, 1/3
Sum of zeroes = √2, Product of zeroes = 1/3
Polynomial: x² - (√2)x + (1/3)
Multiplying by 3: 3x² - 3√2x + 1
(iii) 0, √5
Sum of zeroes = 0, Product of zeroes = √5
Polynomial: x² - (0)x + (√5)
= x² + √5
(iv) 1, 1
Sum of zeroes = 1, Product of zeroes = 1
Polynomial: x² - (1)x + (1)
= x² - x + 1
(v) -1/4, 1/4
Sum of zeroes = -1/4, Product of zeroes = 1/4
Polynomial: x² - (-1/4)x + (1/4)
= x² + (1/4)x + (1/4)
Multiplying by 4: 4x² + x + 1
(vi) 4, 1
Sum of zeroes = 4, Product of zeroes = 1
Polynomial: x² - (4)x + (1)
= x² - 4x + 1
Key Concepts
Types of Polynomials
- Linear Polynomial: Degree 1 (e.g., ax + b)
- Quadratic Polynomial: Degree 2 (e.g., ax² + bx + c)
- Cubic Polynomial: Degree 3 (e.g., ax³ + bx² + cx + d)
Zeroes of a Polynomial
A real number k is a zero of polynomial p(x) if p(k) = 0.
Geometrically, zeroes are the x-coordinates of points where the graph of y = p(x) intersects the x-axis.
Relationship Between Zeroes and Coefficients
Maximum Number of Zeroes
- A polynomial of degree n has at most n zeroes
- A quadratic polynomial can have at most 2 zeroes
- A cubic polynomial can have at most 3 zeroes
Geometrical Meaning of Zeroes
- For a linear polynomial: Graph is a straight line intersecting x-axis at one point
- For a quadratic polynomial: Graph is a parabola that can intersect x-axis at 0, 1, or 2 points
- For a cubic polynomial: Graph can intersect x-axis at 1, 2, or 3 points