Tasks 01-04 - Advanced Factorization & Radicals

Mastering Complex Algebraic Techniques

Problem 1: Advanced Factorization - Quadratics

Factor the following quadratic expressions completely:

  1. \(x^2 + 9x + 20\)

  2. \(x^2 - 3x - 18\)

  3. \(2x^2 + 7x + 3\)

  4. \(3x^2 - 11x + 6\)

  5. \(6x^2 + 13x - 5\)

  6. \(4x^2 - 12x + 9\)

  1. \(x^2 + 9x + 20\)
    • Need factors of 20 that sum to 9: (4,5)
    • \((x + 4)(x + 5)\)
  2. \(x^2 - 3x - 18\)
    • Need factors of -18 that sum to -3: (3,-6)
    • \((x + 3)(x - 6)\)
  3. \(2x^2 + 7x + 3\)
    • AC = 6, need factors that sum to 7: (1,6)
    • \(2x^2 + x + 6x + 3 = x(2x + 1) + 3(2x + 1)\)
    • \((x + 3)(2x + 1)\)
  4. \(3x^2 - 11x + 6\)
    • AC = 18, need factors that sum to -11: (-2,-9)
    • \(3x^2 - 2x - 9x + 6 = x(3x - 2) - 3(3x - 2)\)
    • \((x - 3)(3x - 2)\)
  5. \(6x^2 + 13x - 5\)
    • AC = -30, need factors that sum to 13: (15,-2)
    • \(6x^2 + 15x - 2x - 5 = 3x(2x + 5) - 1(2x + 5)\)
    • \((3x - 1)(2x + 5)\)
  6. \(4x^2 - 12x + 9\)
    • Perfect square: \((2x - 3)^2\)

Problem 2: Factoring by Grouping & Cubes

Factor the following expressions completely:

  1. \(x^3 + 3x^2 - 4x - 12\)

  2. \(2x^3 - 5x^2 - 8x + 20\)

  3. \(x^3 - 125\)

  4. \(8x^3 + 27\)

  5. \(64x^3 - 1\)

  6. \(x^3 + 4x^2 - 9x - 36\)

  1. \(x^3 + 3x^2 - 4x - 12\)
    • Group: \((x^3 + 3x^2) + (-4x - 12)\)
    • \(= x^2(x + 3) - 4(x + 3)\)
    • \(= (x + 3)(x^2 - 4)\)
    • \(= (x + 3)(x + 2)(x - 2)\)
  2. \(2x^3 - 5x^2 - 8x + 20\)
    • Group: \((2x^3 - 5x^2) + (-8x + 20)\)
    • \(= x^2(2x - 5) - 4(2x - 5)\)
    • \(= (2x - 5)(x^2 - 4)\)
    • \(= (2x - 5)(x + 2)(x - 2)\)
  3. \(x^3 - 125\)
    • Difference of cubes: \(x^3 - 5^3\)
    • \(= (x - 5)(x^2 + 5x + 25)\)
  4. \(8x^3 + 27\)
    • Sum of cubes: \((2x)^3 + 3^3\)
    • \(= (2x + 3)(4x^2 - 6x + 9)\)
  5. \(64x^3 - 1\)
    • Difference of cubes: \((4x)^3 - 1^3\)
    • \(= (4x - 1)(16x^2 + 4x + 1)\)
  6. \(x^3 + 4x^2 - 9x - 36\)
    • Group: \((x^3 + 4x^2) + (-9x - 36)\)
    • \(= x^2(x + 4) - 9(x + 4)\)
    • \(= (x + 4)(x^2 - 9)\)
    • \(= (x + 4)(x + 3)(x - 3)\)

Problem 3: Simplifying Radicals

Simplify the following radical expressions:

  1. \(\sqrt{72}\)

  2. \(\sqrt{108x^5y^3}\)

  3. \(\sqrt[3]{54a^7b^4}\)

  4. \(3\sqrt{50} + 2\sqrt{32} - \sqrt{200}\)

  5. \(\sqrt{45x^3} - x\sqrt{20x} + 2\sqrt{80x^3}\)

  6. \(\frac{\sqrt{48x^5}}{\sqrt{3x}}\)

  1. \(\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}\)

  2. \(\sqrt{108x^5y^3}\)

    • \(= \sqrt{36 \times 3 \times x^4 \times x \times y^2 \times y}\)
    • \(= 6x^2y\sqrt{3xy}\)

  3. \(\sqrt[3]{54a^7b^4}\)

    • \(= \sqrt[3]{27 \times 2 \times a^6 \times a \times b^3 \times b}\)
    • \(= 3a^2b\sqrt[3]{2ab}\)

  4. \(3\sqrt{50} + 2\sqrt{32} - \sqrt{200}\)

    • \(= 3 \times 5\sqrt{2} + 2 \times 4\sqrt{2} - 10\sqrt{2}\)
    • \(= 15\sqrt{2} + 8\sqrt{2} - 10\sqrt{2}\)
    • \(= 13\sqrt{2}\)

  5. \(\sqrt{45x^3} - x\sqrt{20x} + 2\sqrt{80x^3}\)

    • \(= 3x\sqrt{5x} - x \times 2\sqrt{5x} + 2 \times 4x\sqrt{5x}\)
    • \(= 3x\sqrt{5x} - 2x\sqrt{5x} + 8x\sqrt{5x}\)
    • \(= 9x\sqrt{5x}\)

  6. \(\frac{\sqrt{48x^5}}{\sqrt{3x}} = \sqrt{\frac{48x^5}{3x}} = \sqrt{16x^4} = 4x^2\)

Problem 4: Rationalizing Denominators

Rationalize the following expressions:

  1. \(\frac{5}{\sqrt{3}}\)

  2. \(\frac{6}{\sqrt[3]{9}}\)

  3. \(\frac{3}{\sqrt{7} - 2}\)

  4. \(\frac{4}{3 + \sqrt{5}}\)

  5. \(\frac{\sqrt{6} - \sqrt{2}}{\sqrt{6} + \sqrt{2}}\)

  6. \(\frac{2\sqrt{3}}{2\sqrt{3} - 3\sqrt{2}}\)

  1. \(\frac{5}{\sqrt{3}} = \frac{5\sqrt{3}}{3}\)

  2. \(\frac{6}{\sqrt[3]{9}} = \frac{6}{\sqrt[3]{9}} \times \frac{\sqrt[3]{3}}{\sqrt[3]{3}} = \frac{6\sqrt[3]{3}}{3} = 2\sqrt[3]{3}\)

  3. \(\frac{3}{\sqrt{7} - 2}\)

    • Multiply by \(\frac{\sqrt{7} + 2}{\sqrt{7} + 2}\)
    • \(= \frac{3(\sqrt{7} + 2)}{7 - 4} = \frac{3\sqrt{7} + 6}{3} = \sqrt{7} + 2\)

  4. \(\frac{4}{3 + \sqrt{5}}\)

    • Multiply by \(\frac{3 - \sqrt{5}}{3 - \sqrt{5}}\)
    • \(= \frac{4(3 - \sqrt{5})}{9 - 5} = \frac{12 - 4\sqrt{5}}{4} = 3 - \sqrt{5}\)

  5. \(\frac{\sqrt{6} - \sqrt{2}}{\sqrt{6} + \sqrt{2}}\)

    • Multiply by \(\frac{\sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}\)
    • Numerator: \((\sqrt{6} - \sqrt{2})^2 = 6 - 2\sqrt{12} + 2 = 8 - 4\sqrt{3}\)
    • Denominator: \(6 - 2 = 4\)
    • Result: \(\frac{8 - 4\sqrt{3}}{4} = 2 - \sqrt{3}\)

  6. \(\frac{2\sqrt{3}}{2\sqrt{3} - 3\sqrt{2}}\)

    • Multiply by \(\frac{2\sqrt{3} + 3\sqrt{2}}{2\sqrt{3} + 3\sqrt{2}}\)
    • Numerator: \(2\sqrt{3}(2\sqrt{3} + 3\sqrt{2}) = 12 + 6\sqrt{6}\)
    • Denominator: \((2\sqrt{3})^2 - (3\sqrt{2})^2 = 12 - 18 = -6\)
    • Result: \(\frac{12 + 6\sqrt{6}}{-6} = -2 - \sqrt{6}\)