Skip to main content
Global

5.5E: Mazoezi

  1. Last updated
  2. Save as PDF
  • Page ID
    176034

    • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Mazoezi hufanya kamili

    Kugawanya Monomials

    Katika mazoezi yafuatayo, ugawanye monomials.

    1. \(15r^4s^9÷(15r^4s^9)\)

    2. \(20m^8n^4÷(30m^5n^9)\)

    Jibu

    \(\dfrac{2m^3}{3n^5}\)

    3. \(\dfrac{18a^4b^8}{−27a^9b^5}\)

    4. \(\dfrac{45x^5y^9}{−60x^8y^6}\)

    Jibu

    \(\dfrac{−3y^3}{4x^3}\)

    5. \(\dfrac{(10m^5n^4)(5m^3n^6)}{25m^7n^5}\)

    6. \(\dfrac{(−18p^4q^7)(−6p^3q^8)}{−36p^{12}q^{10}}\)

    Jibu

    \(\dfrac{−3q^5}{p^5}\)

    7. \(\dfrac{(6a^4b^3)(4ab^5)}{(12a^2b)(a^3b)}\)

    8. \(\dfrac{(4u^2v^5)(15u^3v)}{(12u^3v)(u^4v)}\)

    Jibu

    \(\dfrac{5v^4}{u^2}\)

    Gawanya Polynomial na Monomial

    Katika mazoezi yafuatayo, kugawanya kila polynomial na monomial.

    9. \((9n^4+6n^3)÷3n\)

    10. \((8x^3+6x^2)÷2x\)

    Jibu

    \(4x^2+3x\)

    11. \((63m^4−42m^3)÷(−7m^2)\)

    12. \((48y^4−24y^3)÷(−8y^2)\)

    Jibu

    \(−6y^2+3y\)

    13. \(\dfrac{66x^3y^2−110x^2y^3−44x^4y^3}{11x^2y^2}\)

    14. \(\dfrac{72r^5s^2+132r^4s^3−96r^3s^5}{12r^2s^2}\)

    Jibu

    \(6r^3+11r^2s−8rs^3\)

    15. \(10x^2+5x−4−5x\)

    16. \(20y^2+12y−1−4y\)

    Jibu

    \(−5y−3+\dfrac{1}{4y}\)

    Gawanya Polynomials kwa kutumia Division

    Katika mazoezi yafuatayo, kugawanya kila polynomial na binomial.

    17. \((y^2+7y+12)÷(y+3)\)

    18. \((a^2−2a−35)÷(a+5)\)

    Jibu

    \(a−7\)

    19. \((6m^2−19m−20)÷(m−4)\)

    20. \((4x^2−17x−15)÷(x−5)\)

    Jibu

    \(4x+3\)

    21. \((q^2+2q+20)÷(q+6)\)

    22. \((p^2+11p+16)÷(p+8)\)

    Jibu

    \(p+3−\dfrac{8}{p+8}\)

    23. \((3b^3+b^2+4)÷(b+1)\)

    24. \((2n^3−10n+28)÷(n+3)\)

    Jibu

    \(\dfrac{2n^2−6n+8+4}{n+3}\)

    25. \((z^3+1)÷(z+1)\)

    26. \((m^3+1000)÷(m+10)\)

    Jibu

    \(m^2−10m+100\)

    27. \((64x^3−27)÷(4x−3)\)

    28. \((125y^3−64)÷(5y−4)\)

    Jibu

    \(25y^2+20x+16\)

    Gawanya Polynomials kwa kutumia Division

    Katika mazoezi yafuatayo, tumia Idara ya synthetic ili kupata quotient na salio.

    29. \(x^3−6x^2+5x+14\)imegawanywa na\(x+1\)

    30. \(x^3−3x^2−4x+12\)imegawanywa na\(x+2\)

    Jibu

    \(x^2−5x+6; \space 0\)

    31. \(2x^3−11x^2+11x+12\)imegawanywa na\(x−3\)

    32. \(2x^3−11x^2+16x−12\)imegawanywa na\(x−4\)

    Jibu

    \(2x^2−3x+4; \space 4\)

    33. \(x^4-5x^2+2+13x+3\)imegawanywa na\(x+3\)

    34. \(x^4+x^2+6x−10\)imegawanywa na\(x+2\)

    Jibu

    \(x^3−2x^2+5x−4; \space −2\)

    35. \(2x^4−9x^3+5x^2−3x−6\)imegawanywa na\(x−4\)

    36. \(3x^4−11x^3+2x^2+10x+6\)imegawanywa na\(x−3\)

    Jibu

    \(3x^3−2x^2−4x−2;\space 0\)

    Gawanya Kazi za Polynomial

    Katika mazoezi yafuatayo, ugawanye.

    37. Kwa ajili ya kazi\(f(x)=x^2−13x+36\) na\(g(x)=x−4\), kupata ⓐ\(\left(\dfrac{f}{g}\right)(x)\) ⓑ\(\left(\dfrac{f}{g}\right)(−1)\)

    38. Kwa ajili ya kazi\(f(x)=x^2−15x+54\) na\(g(x)=x−9\), kupata ⓐ\(\left(\dfrac{f}{g}\right)(x)\) ⓑ\(\left(\dfrac{f}{g}\right)(−5)\)

    Jibu

    ⓐ\(\left(\dfrac{f}{g}\right)(x)=x−6\)
    ⓑ\(\left(\dfrac{f}{g}\right)(−5)=−11\)

    39. Kwa ajili ya kazi\(f(x)=x^3+x^2−7x+2\) na\(g(x)=x−2\), kupata ⓐ\(\left(\dfrac{f}{g}\right)(x)\) ⓑ\(\left(\dfrac{f}{g}\right)(2)\)

    40. Kwa ajili ya kazi\(f(x)=x^3+2x^2−19x+12\) na\(g(x)=x−3\), kupata ⓐ\(\left(\dfrac{f}{g}\right)(x)\) ⓑ\(\left(\dfrac{f}{g}\right)(0)\)

    Jibu

    ⓐ\(\left(\dfrac{f}{g}\right)(x)=x^2+5x−4\)
    ⓑ\(\left(\dfrac{f}{g}\right)(0)=−4\)

    41. Kwa ajili ya kazi\(f(x)=x^2−5x+2\) na\(g(x)=x^2−3x−1\), kupata ⓐ\((f·g)(x)\) ⓑ\((f·g)(−1)\)

    42. Kwa ajili ya kazi\(f(x)=x^2+4x−3\) na\(g(x)=x^2+2x+4\), kupata ⓐ\((f·g)(x)\) ⓑ\((f·g)(1)\)

    Jibu

    ⓐ\((f·g)(x)=x^4+6x^3+9x^2+10x−12\); ⓑ\((f·g)(1)=14\)

    Tumia Theorem ya Salio na Sababu

    Katika mazoezi yafuatayo, tumia Theorem ya Salio ili kupata salio.

    43. \(f(x)=x^3−8x+7\)imegawanywa na\(x+3\)

    44. \(f(x)=x^3−4x−9\)imegawanywa na\(x+2\)

    Jibu

    \(−9\)

    45. \(f(x)=2x^3−6x−24\)kugawanywa na\(x−3\)

    46. \(f(x)=7x^2−5x−8\)kugawanywa na\(x−1\)

    Jibu

    \(−6\)

    Katika mazoezi yafuatayo, tumia Theorem ya Factor kuamua kama x-cx-c ni sababu ya kazi ya polynomial.

    47. Kuamua\(x+3\) kama sababu ya\(x^3+8x^2+21x+18\)

    48. Kuamua\(x+4\) kama sababu ya\(x^3+x^2−14x+8\)

    Jibu

    hapana

    49. Kuamua\(x−2\) kama sababu ya\(x^3−7x^2+7x−6\)

    50. Kuamua\(x−3\) kama sababu ya\(x^3−7x^2+11x+3\)

    Jibu

    ndiyo

    Mazoezi ya kuandika

    51. James\(48y+6\) hugawanyika kwa njia\(6\) hii:\(\dfrac{48y+6}{6}=48y\). Ni nini kibaya na hoja zake?

    52. Gawanya\(\dfrac{10x^2+x−12}{2x}\) na ueleze kwa maneno jinsi unavyopata kila neno la quotient.

    Jibu

    Jibu litatofautiana

    53. Eleza wakati unaweza kutumia mgawanyiko wa synthetic.

    54. Kwa maneno yako mwenyewe, weka hatua za mgawanyiko wa synthetic kwa\(x^2+5x+6\) kugawanywa na\(x−2\).

    Jibu

    Majibu yatatofautiana.

    Self Check

    Baada ya kukamilisha mazoezi, tumia orodha hii ili kutathmini ujuzi wako wa malengo ya sehemu hii

    Takwimu inaonyesha meza yenye safu saba na nguzo nne. Mstari wa kwanza ni mstari wa kichwa na huandika kila safu. Kichwa cha kwanza cha safu ni “Naweza...”, pili ni “kwa ujasiri”, ya tatu ni “kwa msaada fulani”, “hakuna minus siipate!”. Chini ya safu ya kwanza ni maneno “kugawanya monomials”, “ugawanye polynomial kwa kutumia monomia”, “ugawanye polynomials kwa kutumia mgawanyiko mrefu”, “ugawanye polynomials kwa kutumia mgawanyiko wa maandishi”, “ugawanye kazi za polynomial”, na “tumia Theorem ya Salio na Factor”. Chini ya nguzo ya pili, ya tatu, ya nne ni nafasi tupu ambapo mwanafunzi anaweza kuangalia kiwango gani cha ustadi waliyopata.

    b Kwa kiwango cha 1-10, ungewezaje kupima ujuzi wako wa sehemu hii kwa kuzingatia majibu yako kwenye orodha? Unawezaje kuboresha hii?