Math topic中文

Sum difference cubes

use of:

  • a3+b3=(a+b)(a2−ab+b2)a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})
  • a3−b3=(a−b)(a2+ab+b2)a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})

Practice questions

  1. 1Show that the equation 3e2x+7ex=63e^{2x}+7e^{x}=6 has only one solution and find its value correct to 2 significant figures. Fac…
  2. 2Factorise 64−(x+1)364 - (x + 1)^3. Hence, solve 64−(x+1)3=15(3−x)64 - (x + 1)^3 = 15(3 - x), expressing non-integer roots in surd form.
  3. 3Factorise x3−125x^3 - 125. The expression −2x3+7x2+ax+b-2x^3 + 7x^2 + ax + b, where aa and bb are integers, has a factor of x−5x - 5 a…
  4. 4若a<0a< 0，b<0b< 0，则p=b2a+a2bp=\dfrac{{b}^{2}}{a}+\dfrac{{a}^{2}}{b} 与 q=a+bq=a+b 的大小关系为 ()(\quad) A. p<qp< q B. p⩽qp\leqslant q…
  5. 5Factorise sin⁡3x+cos⁡3x\sin^3 x + \cos^3 x completely. Show that sin⁡3x+cos⁡3xsin⁡x+cos⁡x=1−12sin⁡2x\frac{\sin^3 x + \cos^3 x}{\sin x + \cos x} = 1 - \frac{1}{2}\sin 2x…
  6. 6Factorise a3+b3a^3 + b^3. Show that sin⁡3x+cos⁡3xsin⁡x+cos⁡x=1−sin⁡2x2\frac{\sin^3 x + \cos^3 x}{\sin x + \cos x} = 1 - \frac{\sin 2x}{2}. Hence, solve the…
  7. 7Factorise (2x+1)3−(2x−1)3\left(2x+1\right)^3 - \left(2x-1\right)^3 completely. Hence show that the equation (2x+1)[(2x+1)3−(2x−1)3]=0\left(2x+1\right)\bigl[\left(2x+1\right)^3 - \left(2x-1\right)^3\bigr]=0…
  8. 8Factorise 3x3−24y33x^3 - 24y^3 completely. Express 7x2+19x+15(x+1)2(x+2)\frac{7x^2 + 19x + 15}{(x + 1)^2(x + 2)} as partial fractions.