Chapter 2: Polynomials

Exercise 2.1 - Detailed Solutions

Exercise 2.1 - Step-by-Step Solutions

❓Question 1: Which are polynomials in one variable?

(i) 4x2−3x+74x^2 - 3x + 7

Check each term:

  • 4x24x^2: power = 2 (whole number) ✓
  • −3x-3x: power = 1 (whole number) ✓
  • 77: power = 0 (whole number) ✓

Answer: YES, polynomial in xx ✓

(ii) y2+2y^2 + \sqrt{2}

Check:

  • y2y^2: power = 2 ✓
  • 2\sqrt{2}: This is just a constant number (≈ 1.414), NOT y\sqrt{y}!

Answer: YES, polynomial in y ✓

(iii) 3t+t23\sqrt{t} + t\sqrt{2}

Check each term:

  • 3t=3t1/23\sqrt{t} = 3t^{1/2}: power = 1/2 (NOT whole number) ✗

Answer: NO ✗ (fractional exponent on variable)

(iv) y+2yy + \frac{2}{y}

Check:

  • yy: power = 1 ✓
  • 2y=2y−1\frac{2}{y} = 2y^{-1}: power = -1 (negative!) ✗

Answer: NO ✗ (negative exponent = variable in denominator)

❓Question 2: Write coefficient of x2x^2

(i) 2+x2+x2 + x^2 + x

Rearrange: x2+x+2x^2 + x + 2 Coefficient of x2x^2 = 1 (understood coefficient)

(ii) 2−x2+x32 - x^2 + x^3

Rearrange: x3−x2+2x^3 - x^2 + 2 Coefficient of x2x^2 = -1

❓Question 4: Degree of each polynomial

(i) 5x3+4x2+7x5x^3 + 4x^2 + 7x

Highest power = 3 Degree = 3 (Cubic)

(ii) 4−y24 - y^2

Rearrange: −y2+4-y^2 + 4 Highest power = 2 Degree = 2 (Quadratic)

(iii) 5t−75t - \sqrt{7}

Highest power = 1 Degree = 1 (Linear)

(iv) 33

No variable at all! Degree = 0 (Constant)