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12.5E：练习

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    练习成就完美

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

    在以下练习中，使用 Pascal 的三角形展开每个二项式。

    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. \((3 x+5 y)^{3}\)
    回答

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

    练习\(\PageIndex{20}\) 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. \(\left( \begin{array}{l}{11} \\ {9}\end{array}\right)\)
    回答

    2。

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

    4。

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

    在以下练习中，展开每个二项式。

    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. \((3x+4y)^{5}\)
    回答

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

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

    在以下练习中，在二项式扩展中找到指定的术语。

    1. 的第六学期\((x+y)^{10}\)
    2. 的第五个任期\((a+b)^{9}\)
    3. 第四学期\((x-y)^{8}\)
    4. 的第七个任期\((x-y)^{11}\)
    回答

    2。 \(126a^{5} b^{4}\)

    4。 \(462x^{5} y^{6}\)

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

    在以下练习中，在二项式展开中找到指定项的系数。

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

    2。 \(112\)

    4。 \(324\)

    6。 \(30,618\)

    练习\(\PageIndex{24}\) Writing Exercises
    1. 用你自己的话解释如何找到帕斯卡三角形的行。 写下帕斯卡三角形的前五行。
    2. 用你自己的话说，解释扩展中每个变量的指数模式。
    3. 用你自己的话解释\((a+b)^{n}\)和之间的区别\((a-b)^{n}\)。
    4. 用你自己的话说，解释一下如何在不扩展整个事物的情况下在二项式的扩展中找到一个特定的术语。 用一个例子来帮助解释。
    回答

    2。 答案会有所不同

    4。 答案会有所不同

    自检

    a. 完成练习后，使用此清单评估您对本节目标的掌握程度。

    此图显示了一个包含四行四列的表。 第一行是标题行，读取。 “我可以”、“自信”、“有帮助” 和 “不，我不明白”。 第一列从第二行开始是 “使用帕斯卡三角形展开二项式”、“计算二项式系数” 和 “使用二项式定理展开二项式”。 其余列为空白。
    图 12.4.31

    b. 在 1-10 分中，根据你在清单上的回答，你会如何评价你对本节的掌握程度？ 你怎么能改善这个？

    This page titled 12.5E：练习 is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.