Chapter 2: Polynomials — notes

1. 1. Polynomials and their degree

   First, a quick recall.

   • • Polynomial in x: every power of x is a whole number
   • • Names: p(x), q(x), r(x)
   • • Degree = the highest power of x
   • • 4x + 2 → highest power 1 → degree 1
   • • 2y² − 3y + 4 → degree 2
   • • 5x³ − 4x² + x − √2 → degree 3
   • • 7 = 7x⁰ → degree 0
   • • 0 = 0x = 0x² = 0x⁵ … → degree not defined
   • • Degree 1: linear polynomial, ax + b, a ≠ 0
   • • Example: 2x − 3
   • • Degree 2: quadratic polynomial, ax² + bx + c, a ≠ 0
   • • Example: x² − 3x − 4
   • • Degree 3: cubic polynomial, ax³ + bx² + cx + d, a ≠ 0
   • • Example: x³ − 4x
   • • | Name | Degree | General form |
   • • | Linear | 1 | ax + b |
   • • | Quadratic | 2 | ax² + bx + c |
   • • | Cubic | 3 | ax³ + bx² + cx + d |

2. 2. Zero of a polynomial

   Take p(x) = x² − 3x − 4.

   • • p(x) = x² − 3x − 4
   • • p(2) = 2² − 3 × 2 − 4
   • • = 4 − 6 − 4 = −6
   • • k is a zero of p(x) if p(k) = 0
   • • p(2) = −6 ≠ 0, so 2 is not a zero
   • • p(−1) = (−1)² − 3 × (−1) − 4
   • • = 1 + 3 − 4
   • • = 0, so −1 is a zero
   • • p(4) = 4² − 3 × 4 − 4
   • • = 16 − 12 − 4 = 0, so 4 is a zero
   • • Zeroes of x² − 3x − 4: −1 and 4
   • • Zero of ax + b
   • • ak + b = 0
   • • ak = −b
   • • k = (−b)/a
   • • 2x + 3: zero = −3/2
   • • Check: 2 × (−3/2) + 3 = −3 + 3 = 0 ✓

3. 3. Finding zeroes: splitting the middle term

   We do not guess.

   • • Splitting the middle term of ax² + bx + c
   • • Find two numbers: product = a × c, sum = b
   • • x² − 3x − 4: a = 1, b = −3, c = −4
   • • a × c = 1 × (−4) = −4
   • • −4 × 1 = −4, −4 + 1 = −3
   • • x² − 3x − 4 = x² − 4x + x − 4
   • • = x(x − 4) + 1(x − 4)
   • • = (x − 4)(x + 1)
   • • (x − 4)(x + 1) = 0
   • • x − 4 = 0 ⇒ x = 4
   • • x + 1 = 0 ⇒ x = −1
   • • Zeroes: 4 and −1
   • • 2x² − 8x + 6
   • • a × c = 2 × 6 = 12
   • • (−6) × (−2) = 12, −6 + (−2) = −8
   • • 2x² − 8x + 6 = 2x² − 6x − 2x + 6
   • • = 2x(x − 3) − 2(x − 3)
   • • = (2x − 2)(x − 3)
   • • = 2(x − 1)(x − 3)
   • • x − 1 = 0 ⇒ x = 1, x − 3 = 0 ⇒ x = 3
   • • Zeroes of 2x² − 8x + 6: 1 and 3

4. 4. Geometrical meaning of the zeroes

   Take y = 2x + 3.

   • • y = 2x + 3
   • • x = −2 ⇒ y = 2 × (−2) + 3 = −4 + 3 = −1
   • • x = 0 ⇒ y = 2 × 0 + 3 = 3
   • • | x | −2 | 0 |
   • • | y | −1 | 3 |
   • • Line through (−2, −1) and (0, 3)
   • • On the x-axis, y = 0: 2x + 3 = 0 ⇒ x = −3/2
   • • Cuts the x-axis at (−3/2, 0)
   • • Zeroes of p(x) = x-coordinates where the graph of y = p(x) meets the x-axis
   • • Linear polynomial: a line, meets the x-axis once → exactly 1 zero
   • • y = x² − 3x − 4
   • • x = −2 ⇒ y = 4 + 6 − 4 = 6
   • • x = −1 ⇒ y = 1 + 3 − 4 = 0
   • • x = 0 ⇒ y = 0 − 0 − 4 = −4
   • • x = 1 ⇒ y = 1 − 3 − 4 = −6
   • • x = 2 ⇒ y = 4 − 6 − 4 = −6
   • • x = 3 ⇒ y = 9 − 9 − 4 = −4
   • • x = 4 ⇒ y = 16 − 12 − 4 = 0
   • • x = 5 ⇒ y = 25 − 15 − 4 = 6
   • • | x | −2 | −1 | 0 | 1 | 2 | 3 | 4 | 5 |
   • • | y | 6 | 0 | −4 | −6 | −6 | −4 | 0 | 6 |
   • • Parabola meets the x-axis at (−1, 0) and (4, 0) → zeroes −1, 4
   • • a > 0 → parabola opens upwards ∪
   • • a < 0 → parabola opens downwards ∩
   • • Quadratic: three cases
   • • Case 1: cuts the x-axis at 2 points → 2 zeroes
   • • Case 2: touches at 1 point → 2 equal (coincident) zeroes
   • • Case 3: does not meet the x-axis → no zero
   • • So a quadratic has at most 2 zeroes
   • • Cubic: x³ − 4x
   • • x³ − 4x = x(x² − 4)
   • • = x(x − 2)(x + 2)
   • • Zeroes: −2, 0, 2 → graph cuts the x-axis 3 times
   • • x³ → zero: 0 (1 zero)
   • • x³ − x² = x²(x − 1) → zeroes: 0, 1 (2 zeroes)
   • • Cubic: at most 3 zeroes
   • • A polynomial of degree n has at most n zeroes

5. 5. Zeroes and coefficients of a quadratic

   Something beautiful.

   • • 2x² − 8x + 6: zeroes 1 and 3
   • • Sum of zeroes = 1 + 3 = 4
   • • (−(−8))/2 = 8/2 = 4
   • • Product of zeroes = 1 × 3 = 3
   • • 6/2 = 3
   • • Why it works
   • • ax² + bx + c = k(x − α)(x − β)
   • • = kx² − k(α + β)x + kαβ
   • • a = k
   • • b = −k(α + β) ⇒ α + β = (−b)/a
   • • c = kαβ ⇒ αβ = c/a
   • • Sum of zeroes: α + β = (−b)/a = (−(coefficient of x))/(coefficient of x²)
   • • Product of zeroes: αβ = c/a = (constant term)/(coefficient of x²)
   • • x² − 3: a = 1, b = 0, c = −3
   • • x² − 3 = (x − √3)(x + √3) ⇒ zeroes √3, −√3
   • • Sum = √3 + (−√3) = 0, (−b)/a = (−0)/1 = 0
   • • Product = √3 × (−√3) = −3, c/a = (−3)/1 = −3

6. 6. A quadratic from its zeroes

   We can go backwards too.

   • • Polynomial from its zeroes
   • • k(x − α)(x − β) = k(x² − (α + β)x + αβ)
   • • k[x² − (sum of zeroes)x + (product of zeroes)], k ≠ 0
   • • Sum = −3, product = 2, k = 1
   • • x² − (−3)x + 2
   • • = x² + 3x + 2
   • • x² + 3x + 2 = (x + 1)(x + 2) ⇒ zeroes −1, −2
   • • Check: −1 + (−2) = −3 ✓, (−1) × (−2) = 2 ✓
   • • Sum = 1/4, product = −1
   • • x² − 1/4x − 1
   • • k = 4: 4(x² − 1/4x − 1) = 4x² − x − 4

7. 7. Zeroes and coefficients of a cubic

   A cubic has up to three zeroes, so it has three relations.

   • • Cubic ax³ + bx² + cx + d, zeroes α, β, γ
   • • α + β + γ = (−b)/a
   • • αβ + βγ + γα = c/a
   • • αβγ = (−d)/a
   • • p(x) = 2x³ − 5x² − 14x + 8, zeroes 4, −2, 1/2
   • • a = 2, b = −5, c = −14, d = 8
   • • p(4) = 2 × 64 − 5 × 16 − 14 × 4 + 8
   • • = 128 − 80 − 56 + 8 = 0 ✓
   • • α + β + γ = 4 + (−2) + 1/2 = 5/2
   • • (−b)/a = (−(−5))/2 = 5/2 ✓
   • • αβ + βγ + γα = 4 × (−2) + (−2) × 1/2 + 1/2 × 4
   • • = −8 − 1 + 2 = −7
   • • c/a = (−14)/2 = −7 ✓
   • • αβγ = 4 × (−2) × 1/2 = −4
   • • (−d)/a = (−8)/2 = −4 ✓

AskMedha · Medha Ma’am — CBSE Class 10 Maths, Chapter 2. Written from the textbook, checked twice.