Solving Polynomials by Hand Solutions

Author: Indigo Curnick

Date: 2025-04-02

#mathematics  


These are the solutions to the Solving Polynomials by Hand blog. Don't look at these till you try all the problems!

Factorising by Inspection

  1. \((x+2)(x-5)\)
  2. \((x-1)^2 = (x-1)(x-1)\)
  3. \((x+3)(2x-1)\)
  4. \((3x+1)(2x-4)\)

Quadratic Formula

  1. \(x = - \frac{3}{2} - \frac{\sqrt{7}}{2} \approx -2.8229, x = - \frac{3}{2} + \frac{\sqrt{7}}{2} \approx -0.1771 \)
  2. \(x = - \frac{1}{8} - \frac{\sqrt{113}}{8} \approx -1.4538, x = \frac{\sqrt{113}}{8} - \frac{1}{8} \approx 1.2308\)
  3. \(x = 1 - \sqrt{10} \approx -2.1623, x = 1 + \sqrt{10} \approx 4.1623 \)
  4. \(x = \frac{25}{14} - \frac{\sqrt{559}}{14} \approx 0.0819, x = \frac{25}{14} + \frac{\sqrt{569}}{14} \approx 3.4896\)

Completing the Square

  1. \( (x+6)^2 - 37 \)
  2. \( (x+3)^2 + 16 \)
  3. \( 3(x-2)^2 - 8 \)
  4. \( 3(x+3)^2 - 28 \)

Comparing Coefficients

  1. \( (x-2)(x-1)(x+3) \)
  2. \( (x-4)(x+1)(2x-3) \)
  3. \( (x-1)(x+1)(x+2)(x-3) \)
  4. \( (2x+1)(3x-1)(x+1)(x-3)(x-1) \)