แบบฝึกหัด การแยกตัวประกอบของพหุนามดีกรีสอง

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1. จงแยกตัวประกอบของพหุนามต่อไปนี้

  1. \(2x^2 + 5x + 2\)
  2. \(2x^2 + x \; – 6\)
  3. \(3x^2 \;- 13x + 12\)
  4. \(6x^2 + 5x \; – 25\)
  5. \(3x^2 \; – x \; – 10\)
  6. \(8x^2 + 2x \; – 15\)
  7. \(4x^2 + 19x \; – 30\)
  8. \(7x^2 + 39x \; – 18\)
  9. \(5x^2 + 23x + 12\)
  10. \(9x^2 + 9x \; – 40\)
  11. \(6x^2 + 19x + 15\)
  12. \(10x^2 + 29x \; – 21\)
  13. \(2x^2 \; – 5x \; – 3\)
  14. \(-5x^2 + 7x + 6\)
  15. \(2x^2 \; – 7x \; – 15\)
  16. \(-6x^2 +13x + 8\)
  17. \(3x^2 \; – 17x \; – 6\)
  18. \(-18x^2 \; – 9x + 2\)
  19. \(6x^2 \; – x \; – 40\)
  20. \(16x^2 + 66x \; – 27\)
  21. \(7x^2 \; – 5x \; – 2\)
  22. \(28x^2 + 31x \; – 5\)
  23. \(8x^2 \; – 6x \; – 9\)
  24. \(12x^2 \; – 29x + 14\)
ดูเฉลยคำตอบ
  1. \(2x^2 + 5x + 2 = (2x + 1)(x + 2)\)
  2. \(2x^2 + x \; – 6 = (2x \; – 3)(x + 2)\)
  3. \(3x^2 \;- 13x + 12 = (3x \; – 4)(x \; – 3)\)
  4. \(6x^2 + 5x \; – 25 = (3x \; – 5)(2x + 5)\)
  5. \(3x^2 \; – x \; – 10 = (3x + 5)(x \; – 2)\)
  6. \(8x^2 + 2x \; – 15 = (4x \; – 5)(2x + 3)\)
  7. \(4x^2 + 19x \; – 30 = (4x \; – 5)(x + 6)\)
  8. \(7x^2 + 39x \; – 18 = (7x \; – 3)(x + 6)\)
  9. \(5x^2 + 23x + 12 = (5x + 3)(x + 4)\)
  10. \(9x^2 + 9x \; – 40 = (3x + 8)(3x \; – 5)\)
  11. \(6x^2 + 19x + 15 = (3x + 5)(2x + 3)\)
  12. \(10x^2 + 29x \; – 21 = (5x \; – 3)(2x + 7)\)
  13. \(2x^2 \; – 5x \; – 3 = (2x + 1)(x \; – 3)\)
  14. \(-5x^2 + 7x + 6 = (-5x \; – 3)(x \; – 2)\)
  15. \(2x^2 \; – 7x \; – 15 = (2x + 3)(x \; – 5)\)
  16. \(-6x^2 +13x + 8 = (-3x + 8)(2x + 1)\)
  17. \(3x^2 \; – 17x \; – 6 = (3x + 1)(x \; – 6)\)
  18. \(-18x^2 \; – 9x + 2 = (6x \; – 1)(-3x \; – 2)\)
  19. \(6x^2 \; – x \; – 40 = (3x \; – 8)(2x + 5)\)
  20. \(16x^2 + 66x \; – 27 = (8x \; – 3)(2x + 9)\)
  21. \(7x^2 \; – 5x \; – 2 = (7x + 2)(x \; – 1)\)
  22. \(28x^2 + 31x \; – 5 = (7x \; – 1)(4x + 5)\)
  23. \(8x^2 \; – 6x \; – 9 = (4x + 3)(2x \; – 3)\)
  24. \(12x^2 \; – 29x + 14 = (3x \; – 2)(4x \; – 7)\)

2. จงแยกตัวประกอบของพหุนามต่อไปนี้ โดยวิธีทำเป็นกำลังสองสมบูรณ์

  1. \(x^2 + 8x \; – 84\)
  2. \(x^2 \; – 8x + 15\)
  3. \(x^2 + 8x + 15\)
  4. \(x^2 \; – 24x + 143\)
  5. \(x^2 \; – 2x \; – 8\)
  6. \(x^2 + 10x \; – 7\)
  7. \(x^2 \; – 7x \; – 1\)
  8. \(x^2 + 5x \; – 3\)
  9. \(2x^2 + 3x + 1\)
  10. \(2x^2 + x \; – 3\)
ดูเฉลยคำตอบ
  1. \(x^2 + 8x \; – 84\)
    \(\begin{align*}
    \qquad &= x^2 + 2(4)x + (4)^2 \; – (4)^2 \;- 84 \\
    &= (x + 4)^2 \; – 16 \; – 84 \\
    &= (x + 4)^2 \; – 10^2 \\
    &= (x + 4 + 10)(x + 4 \; – 10) \\
    &= (x + 14)(x \; – 6)
    \end{align*}\)
  2. \(x^2 \; – 8x + 15\)
    \(\begin{align*}
    \qquad &= x^2 \; – 2(4)x + (4)^2 \; – (4)^2 + 15 \\
    &= (x \; – 4)^2 \; – 16 + 15 \\
    &= (x \; – 4)^2 \; – (1)^2 \\
    &= (x \; – 4 + 1)(x \; – 4 \; – 1) \\
    &= (x \; – 3)(x \; – 5)
    \end{align*}\)
  3. \(x^2 + 8x + 15\)
    \(\begin{align*}
    \qquad &= x^2 \; + 2(4)x + (4)^2 \; – (4)^2 + 15 \\
    &= (x + 4)^2 \; – 16 + 15 \\
    &= (x + 4)^2 \; – (1)^2 \\
    &= (x + 4 + 1)(x + 4 \; – 1) \\
    &= (x + 5)(x + 3)
    \end{align*}\)
  4. \(x^2 \; – 24x + 143\)
    \(\begin{align*}
    \qquad &= x^2 \; – 2(12)x + (12)^2 \; – (12)^2 + 143 \\
    &= (x \; – 12)^2 \; – 144 + 143 \\
    &= (x \; – 12)^2 \; – (1)^2 \\
    &= (x \; – 12 + 1)(x \; – 12 \; – 1) \\
    &= (x \; – 11)(x \; – 13)
    \end{align*}\)
  5. \(x^2 \; – 2x \; – 8\)
    \(\begin{align*}
    \qquad &= x^2 \; – 2(1)x + (1)^2 \; – (1)^2 \; – 8 \\
    &= (x \; – 1)^2 \; – 1 \; – 8 \\
    &= (x \; – 1)^2 \; – (3)^2 \\
    &= (x \; – 1 + 3)(x \; – 1 \; – 3) \\
    &= (x + 2)(x \; – 4)
    \end{align*}\)
  6. \(x^2 + 10x \; – 7\)
    \(\begin{align*}
    \qquad &= x^2 + 2(5)x + (5)^2 \; – (5)^2 \; – 7 \\
    &= (x + 5)^2 \; – 25 \; – 7 \\
    &= (x + 5)^2 \; – (\sqrt{32})^2 \\
    &= (x + 5)^2 \; – (4\sqrt{2})^2 \\
    &= (x + 5 + 4\sqrt{2})(x + 5 \; – 4\sqrt{2})
    \end{align*}\)
  7. \(x^2 \; – 7x \; – 1\)
    \(\begin{align*}
    \qquad &= x^2 \; – 2\Bigl(\frac{7}{2}\Bigl)x + \Bigl(\frac{7}{2}\Bigl)^2 \; – \Bigl(\frac{7}{2}\Bigl)^2 \; – 1 \\
    &= \Bigl(x \; – \frac{7}{2}\Bigl)^2 \; – \frac{49}{4} \; – 1 \\
    &= \Bigl(x \; – \frac{7}{2}\Bigl)^2 \; – \frac{53}{4} \\
    &= \Bigl(x \; – \frac{7}{2}\Bigl)^2 \; – \Bigl(\frac{\sqrt{53}}{2}\Bigl)^2 \\
    &= \Bigl(x \; – \frac{7 + \sqrt{53}}{2}\Bigl)\Bigl(x \; – \frac{7 \; – \sqrt{53}}{2}\Bigl)
    \end{align*}\)
  8. \(x^2 + 5x \; – 3\)
    \(\begin{align*}
    \qquad &= x^2 + 2\Bigl(\frac{5}{2}\Bigl)x + \Bigl(\frac{5}{2}\Bigl)^2 \; – \Bigl(\frac{5}{2}\Bigl)^2 \; – 3 \\
    &= \Bigl(x + \frac{5}{2}\Bigl)^2 \; – \frac{25}{4} \; – 3 \\
    &= \Bigl(x + \frac{5}{2}\Bigl)^2 \; – \frac{37}{4} \\
    &= \Bigl(x + \frac{5}{2}\Bigl)^2 \; – \Bigl(\frac{\sqrt{37}}{2}\Bigl)^2 \\
    &= \Bigl(x + \frac{5 + \sqrt{37}}{2}\Bigl)\Bigl(x + \frac{5 \; – \sqrt{37}}{2}\Bigl)
    \end{align*}\)
  9. \(2x^2 + 3x + 1\)
    \(\begin{align*}
    \qquad &= 2\biggl[x^2 + \frac{3}{2}x + \frac{1}{2}\biggl] \\
    &= 2\biggl[x^2 + 2\Bigl(\frac{3}{4}\Bigl)x + \Bigl(\frac{3}{4}\Bigl)^2 \; – \Bigl(\frac{3}{4}\Bigl)^2 + \frac{1}{2}\biggl] \\
    &= 2\biggl[\Bigl(x + \frac{3}{4}\Bigl)^2 \; – \frac{9}{16} + \frac{1}{2}\biggl] \\
    &= 2\biggl[\Bigl(x + \frac{3}{4}\Bigl)^2 \; – \frac{1}{16}\biggl] \\
    &= 2\biggl[\Bigl(x + \frac{3}{4}\Bigl)^2 \; – \Bigl(\frac{1}{4}\Bigl)^2\biggl] \\
    &= 2\biggl[\Bigl(x + \frac{3}{4} + \frac{1}{4}\Bigl)\Bigl(x + \frac{3}{4} \; – \frac{1}{4}\Bigl)\biggl] \\
    &= 2\biggl[(x + 1)\Bigl(x + \frac{1}{2}\Bigl)\biggl] \end{align*}\)
  10. \(2x^2 + x \; – 3\)
    \(\begin{align*}
    \qquad &= 2\biggl[x^2 + \frac{1}{2}x \; – \frac{3}{2}\biggl] \\
    &= 2\biggl[x^2 + 2\Bigl(\frac{1}{4}\Bigl)x + \Bigl(\frac{1}{4}\Bigl)^2 \; – \Bigl(\frac{1}{4}\Bigl)^2 \; – \frac{3}{2}\biggl] \\
    &= 2\biggl[\Bigl(x + \frac{1}{4}\Bigl)^2 \; – \frac{1}{16} \; – \frac{3}{2}\biggl] \\
    &= 2\biggl[\Bigl(x + \frac{1}{4}\Bigl)^2 \; – \frac{25}{16}\biggl] \\
    &= 2\biggl[\Bigl(x + \frac{1}{4}\Bigl)^2 \; – \Bigl(\frac{5}{4}\Bigl)^2\biggl] \\
    &= 2\biggl[\Bigl(x + \frac{1}{4} + \frac{5}{4}\Bigl)\Bigl(x + \frac{1}{4} \; – \frac{5}{4}\Bigl)\biggl] \\
    &= 2\biggl[\Bigl(x + \frac{3}{2}\Bigl)(x \; – 1)\biggl] \end{align*}\)

3. การแยกตัวประกอบรวมทุกแบบ (โจทย์ระคน)

  1. \(6a^2 + 9a\)
  2. \(5a^4 \; – 10a^2\)
  3. \(a^2 \; – 4\)
  4. \(x^3 \; – x\)
  5. \(x^2 \; – 1\)
  6. \(a^3 + 2a^2\)
  7. \(b^2 \; – 100\)
  8. \(a^2 \; – b^2\)
  9. \(x^2 \; – x\)
  10. \(2x^2 + 6x^2\)
  11. \(9c^2 \; – 16d^2\)
  12. \(c^2 \; – 25\)
  13. \(c^2 \; – 5c\)
  14. \(m^2 \; – 4n^2\)
  15. \(a^2b^2 \; – 9\)
  16. \(2a^2 + a\)
  17. \(4a^3 \; – a\)
  18. \(a^3 \; – b^3\)
  19. \(a^2 \; – 2ab + b^2\)
  20. \(a^2 + ab\)
  21. \(c^2 \; – 16\)
  22. \(c^2 \; – 8c + 16\)
  23. \(3c \; – 9\)
  24. \(x^2 + 3x + 2\)
  25. \(x^2 \; – 7x + 6\)
  26. \(x^2 \; – 4x \; – 12\)
  27. \(x^2 + 4x\)
  28. \(x^3 \; – 9x\)
  29. \(x^2 + 2x \; – 8\)
  30. \(x^2 + x \; – 6\)
  31. \(x^2 + 7x + 12\)
  32. \(x^2 + 9x + 20\)
  33. \(x^2 + 3x + 2\)
  34. \(x^2 + 5x + 6\)
  35. \(x^3 + 9x^2 + 20x\)
  36. \(x^2 + 5x + 4\)
  37. \(x^2 + 7x + 10\)
  38. \(x^2 + 3x + 2\)
  39. \(y^2 \; – 2y \; – 24\)
  40. \(y^2 \; – y \; – 12\)
  41. \(y^2 + 6y + 9\)
  42. \(a^2 \; – 4a \; – 21\)
  43. \(a^3 + 27\)
  44. \(m^2 \; – m \; – 12\)
  45. \(m^3 \; – 64\)
  46. \(m^2 + m \; – 6\)
  47. \(2y \; – y^2 \; – 6y^3\)
  48. \(2 \; – 7y + 6y^2\)
  49. \(4y + 12y^2 + 9y^3\)
  50. \(4 \; – 9y^2\)
  51. \(b^4 \; – 27b\)
  52. \(4b^2 \; – 25\)
  53. \(2b^2 + 5b\)
  54. \(2b^2 \; – 11b + 5\)
  55. \(3x^2 \; – 7x + 2\)
ดูเฉลยคำตอบ
  1. \(6a^2 + 9a = 3a(2a + 3)\)
  2. \(5a^4 \; – 10a^2 = 5a^2(a^2 \; – 2)\)
  3. \(a^2 \; – 4 = (a \; – 2)(a + 2)\)
  4. \(x^3 \; – x = x(x^2 \; – 1) = x(x \; – 1)(x + 1)\)
  5. \(x^2 \; – 1 = (x \; – 1)(x + 1)\)
  6. \(a^3 + 2a^2 = a^2(a + 2)\)
  7. \(b^2 \; – 100 = b^2 \; – 10^2 = (b \; – 10)(b + 10)\)
  8. \(a^2 \; – b^2 = (a \; – b)(a + b)\)
  9. \(x^2 \; – x = x(x \; – 1)\)
  10. \(2x^2 + 6x^2 = 8x^2\)
  11. \(9c^2 \; – 16d^2 = (3c \; – 4d)(3c + 4d)\)
  12. \(c^2 \; – 25 = (c \; – 5)(c + 5)\)
  13. \(c^2 \; – 5c = c(c \; – 5)\)
  14. \(m^2 \; – 4n^2 = (m \; – 2n)(m + 2n)\)
  15. \(a^2b^2 \; – 9 = (ab \; – 3)(ab + 3)\)
  16. \(2a^2 + a = a(2a + 1)\)
  17. \(4a^3 \; – a = a(4a^2 \; – 1)\)
  18. \(a^3 \; – b^3 = (a \; – b)(a^2 + ab + b^2)\)
  19. \(a^2 \; – 2ab + b^2 = (a \; – b)^2\)
  20. \(a^2 + ab = a(a + b)\)
  21. \(c^2 \; – 16 = (c \; – 4)(c + 4)\)
  22. \(c^2 \; – 8c + 16 = (c \; – 4)^2\)
  23. \(3c \; – 9 = 3(c \; – 3)\)
  24. \(x^2 + 3x + 2 = (x + 1)(x + 2)\)
  25. \(x^2 \; – 7x + 6 = (x \; – 6)(x \; – 1)\)
  26. \(x^2 \; – 4x \; – 12 = (x \; – 6)(x + 2)\)
  27. \(x^2 + 4x = x(x + 4)\)
  28. \(x^3 \; – 9x = x(x^2 \; – 9) = x(x \; – 3)(x + 3)\)
  29. \(x^2 + 2x \; – 8 = (x \; – 2)(x + 4)\)
  30. \(x^2 + x \; – 6 = (x \; – 2)(x + 3)\)
  31. \(x^2 + 7x + 12 = (x + 3)(x + 4)\)
  32. \(x^2 + 9x + 20 = (x + 4)(x + 5)\)
  33. \(x^2 + 3x + 2 = (x + 2)(x + 1)\)
  34. \(x^2 + 5x + 6 = (x + 3)(x + 2)\)
  35. \(x^3 + 9x^2 + 20x = x(x^2 + 9x + 20) = (x)(x + 4)(x + 5)\)
  36. \(x^2 + 5x + 4 = (x + 4)(x + 1)\)
  37. \(x^2 + 7x + 10 = (x + 5)(x + 2)\)
  38. \(x^2 + 3x + 2 = (x + 2)(x + 1)\)
  39. \(y^2 \; – 2y \; – 24 = (y \; – 6)(y + 4)\)
  40. \(y^2 \; – y \; – 12 = (y \; – 4)(y + 3)\)
  41. \(y^2 + 6y + 9 = (y + 3)(y + 3) = (y + 3)^2\)
  42. \(a^2 \; – 4a \; – 21 = (a \; – 7)(a + 3)\)
  43. \(a^3 + 27 = a^3 + 3^3 = (a + 3)(a^2 \; – 3a + 9)\)
  44. \(m^2 \; – m \; – 12 = (m \; – 4)(m + 3)\)
  45. \(m^3 \; – 64 = m^3 \; – 4^3 = (m \; – 4)(m^2 + 4m + 16)\)
  46. \(m^2 + m \; – 6 = (m \; – 2)(m + 3)\)
  47. \(2y \; – y^2 \; – 6y^3 = (-y)(6y^2 + y \; – 2) = (-y)(2y \; – 1)(3y + 2)\)
  48. \(2 \; – 7y + 6y^2 = 6y^2 \; – 7y + 2 = (3y \; – 2)(2y \; – 1)\)
  49. \(4y + 12y^2 + 9y^3 = y(9y^2 + 12y + 4) = (y)(3y + 2)(3y + 2) = (y)(3y + 2)^2\)
  50. \(4 \; – 9y^2 = (2 \; – 3y)(2 + 3y)\)
  51. \(b^4 \; – 27b = b(b^3 \; – 27) = b(b \; – 3)(b^2 + 3b + 9)\)
  52. \(4b^2 \; – 25 = (2b)^2 \; – 5^2 = (2b \; – 5)(2b + 5)\)
  53. \(2b^2 + 5b = b(2b + 5)\)
  54. \(2b^2 \; – 11b + 5 = (2b \; – 1)(b \; – 5)\)
  55. \(3x^2 \; – 7x + 2 = (3x \; – 1)(x \; – 2)\)

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