Exercises 12 (Binomial expansion)

  1. Expand the following using the binomial expansion:

    1. \((a+2x)^5\)
    2. \(\left(3a-\dfrac{x}{2}\right)^3\)
    3. \(\left(\dfrac{x}{4}-a\right)^4\)

  2. If \(x\) is so small that terms in \(x^5\) and higher may be neglected, show that \[(x - 3)^2(1 + x)^9 \approx 666x^4 + 549x^3 + 271x^2 + 75x + 9.\]

  3. Expand the following to the fourth term using the binomial expansion:

    1. \((1+6x)^3\)
    2. \(\left(1-\dfrac{5x}{2}\right)^{-3}\)
    3. \(\dfrac{1}{(1+3x)^3}\)
    4. \(\sqrt{1-2x}\)

  4. Assuming that \(x\) is so small that terms in \(x^3\) and higher may be ignored, show that \[\frac{1-\frac{x}{2}}{\sqrt{1+\frac{x}{2}}}\approx 1-\frac{3x}{4}+\frac{7x^2}{32}.\]

  5. Assuming \(x\) is small, expand \(\dfrac{\sqrt{1-x}}{\sqrt{1+2x}}\) up to and including the term in \(x^2\).

  6. Using the binomial expansion, evaluate to three decimal places:

    1. \(\sqrt{1.01}\)
    2. \(\sqrt[3]{27.3}\)

  7. Using the binomial approximation, simplify:

    1. \(\dfrac{\sqrt{1+2x}}{(12+4x)^2}\)
    2. \(\dfrac{1}{(3x-2)(1+3x)^{-\frac12}}\)
    3. \(\dfrac{1-6x+9x^2}{\sqrt{1+6x}}\)

  8. Expand the following in:

    1. \((3 + x)^3\)
    2. \((5 + 2x)^3\)
    3. \((2 + x)^4\)
    4. \((2 - x)^4\)
    5. \((2y + x)^5\)
    6. \((2x - 3y)^5\)
    7. \(\left(x-\dfrac{1}{x}\right)^4\)
    8. \(\left(x-\dfrac{2}{x}\right)^5\)

  9. Expand \((2 + x)^5\) and use your expansion to find a) \((2.1)^5\) and b) \((1.9)^5\).

  10. Expand each of the following in ascending powers of \(x\) up to and including the term in \(x^3\):

    1. \((1 + 2x)(1 - x)^{10}\)
    2. \((1 - 3x)(1 + x)^6\)
    3. \((1 + x^2)(1 + 2x)^8\)

Solutions: 1. (i) \(a^5 + 10a^4x + 40a^3x^2 + 80a^2x^3 + 80ax^4 + 32x^5\); (ii) \(27a^3-\frac{27a^2x}{2}+\frac{9ax^2}{4}-\frac{x^3}{8}\); (iii) \(\frac{x^4}{256}-\frac{x^3a}{16}+\frac{3x^2a^2}{8}-xa^3+a^4\); 3. (i) \(1 +18x +108x^2 + 216x^3\); (ii) \(1+\frac{15x}{2}+\frac{75x^2}{2}+ \frac{625x^3}{4}\); (iii) \(1 - 9x + 54x2 - 270x^3\); iv) \(1-x-\frac{x^2}{2}-\frac{x^3}{2}\); 5. \(1-\frac{3x}{2}+\frac{15x^2}{8}\); 6. (i) \(1.005\); (ii) \(3.011\); 7. (i) \(\frac{1}{144}(1+\frac{x}3)\); (ii) \(-\frac12(1+3x)\); (iii) \(1-9x\); 8. (i) \(27 + 27x + 9x^2 + x^3\); (ii) \(125 + 150x + 60x^2 + 8x^3\); (iii) \(16 + 32x + 24x^2 + 8x^3 + x^4\); (iv) \(16 - 32x + 24x^2 - 8x^3 + x^4\); (v) \(32y^5 + 80y^4x + 80y^3x^2 + 40y^2x^3 + 10yx^4 + x^5\); (vi) \(32x^5 - 240x^4y + 720x^3y^2 - 1080x^2y^3 +810xy^4 - 243y^5\); (vii) \(x^4-4x^2+6-\frac{4}{x^2}+\frac{1}{x^4}\); (viii) \(x^5-10x^3+40x-\frac{80}{x}+\frac{80}{x^3}-\frac{32}{x^5}\); 9. \(32 + 80x + 80x^2 + 40x^3 + 10x^4 + x^5\); (a) \(40.84101\); (b) \(24.76099\); 10. (i) \(1 - 8x + 25x^2 - 30x^3\); (ii) \(1 + 3x - 3x^2 - 25x^3\); (iii) \(1 + 16x + 113x^2 + 464x^3\)