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12.5E: Mazoezi

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    175734

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    Mazoezi hufanya kamili

    Zoezi\(\PageIndex{19}\) Use Pascal's Triangle to Expand a Binomial

    Katika mazoezi yafuatayo, panua kila binomial kwa kutumia Triangle ya Pascal.

    1. \((x+y)^{4}\)
    2. \((a+b)^{8}\)
    3. \((m+n)^{10}\)
    4. \((p+q)^{9}\)
    5. \((x-y)^{5}\)
    6. \((a-b)^{6}\)
    7. \((x+4)^{4}\)
    8. \((x+5)^{3}\)
    9. \((y+2)^{5}\)
    10. \((y+1)^{7}\)
    11. \((z-3)^{5}\)
    12. \((z-2)^{6}\)
    13. \((4x-1)^{3}\)
    14. \((3x-1)^{5}\)
    15. \((3 x-4)^{4}\)
    16. \((3 x-5)^{3}\)
    17. \((2 x+3 y)^{3}\)
    18. \((3 x+5 y)^{3}\)
    Jibu

    2. \(\begin{array}{l}{a^{8}+8 a^{7} b+28 a^{6} b^{2}+56 a^{5} b^{3}} {+70 a^{4} b^{4}+56 a^{3} b^{5}+28 a^{2} b^{6}} {+8 a b^{7}+b^{8}}\end{array}\)

    4. \(\begin{array}{l}{p^{9}+9 p^{8} q+36 p^{7} q^{2}+84 p^{6} q^{3}} {+126 p^{5} q^{4}+126 p^{4} q^{5}+84 p^{3} q^{6}} {+36 p^{2} q^{7}+9 p q^{8}+q^{9}}\end{array}\)

    6. \(\begin{array}{l}{a^{6}-6 a^{5} b+15 a^{4} b^{2}-20 a^{3} b^{3}} {+15 a^{2} b^{4}-6 a b^{5}+b^{6}}\end{array}\)

    8. \(x^{3}+15 x^{2}+75 x+125\)

    10. \(\begin{array}{l}{y^{7}+7 y^{6}+21 y^{5}+35 y^{4}+35 y^{3}} {+21 y^{2}+7 y+1}\end{array}\)

    12. \(\begin{array}{l}{z^{6}-12 z^{5}+60 z^{4}-160 z^{3}+240 z^{2}} \\ {-192 z+64}\end{array}\)

    14. \(\begin{array}{l}{243 x^{5}-405 x^{4}+270 x^{3}-90 x^{2}} {+15 x-1}\end{array}\)

    16. \(27 x^{3}-135 x^{2}+225 x-125\)

    18. \(27 x^{3}+135 x^{2} y+225 x y^{2}+125 y^{3}\)

    Zoezi\(\PageIndex{20}\) Evaluate a Binomial Coefficient
      1. \(\left( \begin{array}{l}{8} \\ {1}\end{array}\right)\)
      2. \(\left( \begin{array}{l}{10} \\ {10}\end{array}\right)\)
      3. \(\left( \begin{array}{l}{6} \\ {0}\end{array}\right)\)
      4. \(\left( \begin{array}{l}{9} \\ {3}\end{array}\right)\)
      1. \(\left( \begin{array}{l}{7} \\ {1}\end{array}\right)\)
      2. \(\left( \begin{array}{l}{4} \\ {4}\end{array}\right)\)
      3. \(\left( \begin{array}{l}{3} \\ {0}\end{array}\right)\)
      4. \(\left( \begin{array}{l}{5} \\ {3}\end{array}\right)\)
      1. \(\left( \begin{array}{l}{3} \\ {1}\end{array}\right)\)
      2. \(\left( \begin{array}{l}{9} \\ {9}\end{array}\right)\)
      3. \(\left( \begin{array}{l}{7} \\ {0}\end{array}\right)\)
      4. \(\left( \begin{array}{l}{5} \\ {3}\end{array}\right)\)
      1. \(\left( \begin{array}{l}{4} \\ {1}\end{array}\right)\)
      2. \(\left( \begin{array}{l}{5} \\ {5}\end{array}\right)\)
      3. \(\left( \begin{array}{l}{8} \\ {0}\end{array}\right)\)
      4. \(\left( \begin{array}{l}{11} \\ {9}\end{array}\right)\)
    Jibu

    2.

    1. \(7\)
    2. \(1\)
    3. \(1\)
    4. \(45\)

    4.

    1. \(4\)
    2. \(1\)
    3. \(1\)
    4. \(55\)
    Zoezi\(\PageIndex{21}\) Use the Binomial Theorem to Expand a Binomial

    Katika mazoezi yafuatayo, kupanua kila binomial.

    1. \((x+y)^{3}\)
    2. \((m+n)^{5}\)
    3. \((a+b)^{6}\)
    4. \((s+t)^{7}\)
    5. \((x-2)^{4}\)
    6. \((y-3)^{4}\)
    7. \((p-1)^{5}\)
    8. \((q-4)^{3}\)
    9. \((3x-y)^{5}\)
    10. \((5x-2y)^{4}\)
    11. \((2x+5y)^{4}\)
    12. \((3x+4y)^{5}\)
    Jibu

    2. \(\begin{array}{l}{m^{5}+5 m^{4} n+10 m^{3} n^{2}+10 m^{2} n^{3}} {+5 m n^{4}+n^{5}}\end{array}\)

    4. \(\begin{array}{l}{s^{7}+7 s^{6} t+21 s^{5} t^{2}+35 s^{4} t^{3}} {+35 s^{3} t^{4}+21 s^{2} t^{5}+7 s t^{6}+t^{7}}\end{array}\)

    6. \(y^{4}-12 y^{3}+54 y^{2}-108 y+81\)

    8. \(q^{3}-12 q^{2}+48 q-64\)

    10. \(\begin{array}{l}{625 x^{4}-1000 x^{3} y+600 x^{2} y^{2}} {-160 x y^{3}+16 y^{4}}\end{array}\)

    12. \(\begin{array}{l}{243 x^{5}+1620 x^{4} y+4320 x^{3} y^{2}} {+5760 x^{2} y^{3}+3840 x y^{4}+1024 y^{5}}\end{array}\)

    Zoezi\(\PageIndex{22}\) Use the Binomial Theorem to Expand a Binomial

    Katika mazoezi yafuatayo, tafuta neno lililoonyeshwa katika upanuzi wa binomial.

    1. Muda wa sita wa\((x+y)^{10}\)
    2. Muda wa tano wa\((a+b)^{9}\)
    3. Muda wa nne wa\((x-y)^{8}\)
    4. Muda wa saba wa\((x-y)^{11}\)
    Jibu

    2. \(126a^{5} b^{4}\)

    4. \(462x^{5} y^{6}\)

    Zoezi\(\PageIndex{23}\) Use the Binomial Theorem to Expand a Binomial

    Katika mazoezi yafuatayo, tafuta mgawo wa muda ulioonyeshwa katika upanuzi wa binomial.

    1. \(y^{3}\)muda wa\((y+5)^{4}\)
    2. \(x^{6}\)muda wa\((x+2)^{8}\)
    3. \(x^{5}\)muda wa\((x-4)^{6}\)
    4. \(x^{7}\)muda wa\((x-3)^{9}\)
    5. \(a^{4} b^{2}\)muda wa\((2 a+b)^{6}\)
    6. \(p^{5} q^{4}\)muda wa\((3 p+q)^{9}\)
    Jibu

    2. \(112\)

    4. \(324\)

    6. \(30,618\)

    Zoezi\(\PageIndex{24}\) Writing Exercises
    1. Kwa maneno yako mwenyewe kuelezea jinsi ya kupata safu za Triangle ya Pascal. Andika safu tano za kwanza za Triangle ya Pascal.
    2. Kwa maneno yako mwenyewe, kueleza mfano wa exponents kwa kila variable katika upanuzi wa.
    3. Kwa maneno yako mwenyewe, kueleza tofauti kati\((a+b)^{n}\) na\((a-b)^{n}\).
    4. Kwa maneno yako mwenyewe, kuelezea jinsi ya kupata muda maalum katika upanuzi wa binomial bila kupanua jambo zima. Tumia mfano ili kusaidia kueleza.
    Jibu

    2. Majibu yatatofautiana

    4. Majibu yatatofautiana

    Self Check

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

    Takwimu hii inaonyesha meza yenye safu nne na nguzo nne. Mstari wa kwanza ni mstari wa kichwa na unasoma. “Naweza”, “Kwa ujasiri”, “Kwa msaada fulani” na “Hapana, siipati”. Safu ya kwanza, kuanzia mstari wa pili inasoma, “Tumia Triangle ya Pascal kupanua Binomial”, “Tathmini mgawo wa Binomial” na “Tumia Theorem ya Binomial kupanua Binomial”. Nguzo zilizobaki ni tupu.
    Kielelezo 12.4.31

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