Chapter 2: Polynomials

Exercise 2.1 - Question 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.
Polynomial Graphs
Graph 1: The polynomial does not cross the x-axis at any point.
Number of zeroes = 0
Graph 2: The polynomial crosses the x-axis exactly once.
Number of zeroes = 1
Graph 3: The polynomial crosses the x-axis three times.
Number of zeroes = 3
Graph 4: The polynomial crosses the x-axis twice.
Number of zeroes = 2
Graph 5: The polynomial crosses the x-axis four times.
Number of zeroes = 4
Graph 6: The polynomial crosses the x-axis three times.
Number of zeroes = 3

Exercise 2.2 - Question 1

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients:

div class="question"> (i) Find the zeroes of x² - 2x - 8 = 0 and verify the relationships between zeroes and coefficients.
Step 1: Use the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a

Here, a = 1, b = -2, c = -8
Discriminant: (-2)² - 4 × 1 × (-8) = 4 + 32 = 36
Roots: x = (2 ± 6) / 2
x₁ = 8 / 2 = 4
x₂ = -4 / 2 = -2
Zeroes: 4 and -2
Step 2: Verify
Sum: 4 + (-2) = 2
Formula: -b/a = -(-2)/1 = 2 ✔ Verified
Product: 4 × (-2) = -8
Formula: c/a = -8/1 = -8 ✔ Verified
(ii) Find the zeroes of 4s² - 4s + 1 = 0 and verify relationships.
Step 1:
a = 4, b = -4, c = 1
Discriminant: (-4)² - 4 × 4 × 1 = 16 - 16 = 0
Equal root: s = -b / 2a = 4 / 8 = 0.5
Zeroes: 0.5 (double root)
Step 2: Verify
Sum: 0.5 + 0.5 = 1
Formula: -b/a = 4/4 = 1 ✔ Verified
Product: 0.5 × 0.5 = 0.25
Formula: c/a = 1/4 = 0.25 ✔ Verified
(iii) Find zeroes of 6x² - 7x - 3 = 0 and verify relationships.
Step 1:
a = 6, b = -7, c = -3
Discriminant: (-7)² - 4 × 6 × (-3) = 49 + 72 = 121
Roots: x = (7 ± 11) / 12
x₁ = 18 / 12 = 1.5, x₂ = -4 / 12 = -0.333...
Step 2: Verify
Sum: 1.5 + (-0.333...) = 1.167 ≈ 7/6
Formula: -b/a = 7/6 ≈ 1.167 ✔ Verified
Product: 1.5 × (-0.333...) = -0.5
Formula: c/a = -3/6 = -0.5 ✔ Verified
(iv) Find zeroes of 4u² + 8u = 0 and verify relationships.
Step 1:
Factorized form: 4u(u + 2) = 0
Zeroes: u = 0, u = -2
Step 2: Verify
Sum: 0 + (-2) = -2
Formula: -b/a = -8/4 = -2 ✔ Verified
Product: 0 × (-2) = 0
Formula: c/a = 0/4 = 0 ✔ Verified
(v) Find zeroes of t² - 15 = 0 and verify relationships.
Step 1:
t² = 15 ⇒ t = ±√15
Step 2: Verify
Sum: √15 + (-√15) = 0
Formula: -b/a = -0/1 = 0 ✔ Verified
Product: √15 × (-√15) = -15
Formula: c/a = -15/1 = -15 ✔ Verified
(vi) Find zeroes of 3x² - x - 4 = 0 and verify relationships.
Step 1:
a = 3, b = -1, c = -4
Discriminant: (-1)² - 4 × 3 × (-4) = 1 + 48 = 49
Roots: x = (1 ± 7) / 6
x₁ = 8 / 6 = 1.333..., x₂ = -6 / 6 = -1
Step 2: Verify
Sum: 1.333... + (-1) = 0.333...
Formula: -b/a = 1/3 = 0.333... ✔ Verified
Product: 1.333... × (-1) = -1.333...
Formula: c/a = -4/3 = -1.333... ✔ Verified
(i) Sum = 1/4, Product = -1
Formula: x² - (Sum)x + Product
= x² - (1/4)x - 1
Polynomial: x² - (1/4)x - 1
(ii) Sum = √2, Product = 1/3
Formula: x² - (√2)x + 1/3
Polynomial: x² - √2·x + 1/3
(iii) Sum = 0, Product = √5
Formula: x² - 0·x + √5
= x² + √5
Polynomial: x² + √5
(iv) Sum = 1, Product = 1
Formula: x² - 1·x + 1
Polynomial: x² - x + 1
(v) Sum = -1/4, Product = 1/4
Formula: x² - (-1/4)x + 1/4
= x² + (1/4)x + 1/4
Polynomial: x² + (1/4)x + 1/4
(vi) Sum = 4, Product = 1
Formula: x² - 4x + 1
Polynomial: x² - 4x + 1