14.5E: Exercises
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Practice Makes Perfect
Exercise \(\PageIndex{19}\) Use Pascal's Triangle to Expand a Binomial
In the following exercises, expand each binomial using Pascal’s Triangle.
- \((x+y)^{4}\)
- \((a+b)^{8}\)
- \((m+n)^{10}\)
- \((p+q)^{9}\)
- \((x-y)^{5}\)
- \((a-b)^{6}\)
- \((x+4)^{4}\)
- \((x+5)^{3}\)
- \((y+2)^{5}\)
- \((y+1)^{7}\)
- \((z-3)^{5}\)
- \((z-2)^{6}\)
- \((4x-1)^{3}\)
- \((3x-1)^{5}\)
- \((3 x-4)^{4}\)
- \((3 x-5)^{3}\)
- \((2 x+3 y)^{3}\)
- \((3 x+5 y)^{3}\)
- Answer
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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. \(27 x^{3}+135 x^{2} y+225 x y^{2}+125 y^{3}\)
Exercise \(\PageIndex{20}\) Evaluate a Binomial Coefficient
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- \(\left( \begin{array}{l}{8} \\ {1}\end{array}\right)\)
- \(\left( \begin{array}{l}{10} \\ {10}\end{array}\right)\)
- \(\left( \begin{array}{l}{6} \\ {0}\end{array}\right)\)
- \(\left( \begin{array}{l}{9} \\ {3}\end{array}\right)\)
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- \(\left( \begin{array}{l}{7} \\ {1}\end{array}\right)\)
- \(\left( \begin{array}{l}{4} \\ {4}\end{array}\right)\)
- \(\left( \begin{array}{l}{3} \\ {0}\end{array}\right)\)
- \(\left( \begin{array}{l}{5} \\ {3}\end{array}\right)\)
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- \(\left( \begin{array}{l}{3} \\ {1}\end{array}\right)\)
- \(\left( \begin{array}{l}{9} \\ {9}\end{array}\right)\)
- \(\left( \begin{array}{l}{7} \\ {0}\end{array}\right)\)
- \(\left( \begin{array}{l}{5} \\ {3}\end{array}\right)\)
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- \(\left( \begin{array}{l}{4} \\ {1}\end{array}\right)\)
- \(\left( \begin{array}{l}{5} \\ {5}\end{array}\right)\)
- \(\left( \begin{array}{l}{8} \\ {0}\end{array}\right)\)
- \(\left( \begin{array}{l}{11} \\ {9}\end{array}\right)\)
- Answer
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2.
- \(7\)
- \(1\)
- \(1\)
- \(45\)
4.
- \(4\)
- \(1\)
- \(1\)
- \(55\)
Exercise \(\PageIndex{21}\) Use the Binomial Theorem to Expand a Binomial
In the following exercises, expand each binomial.
- \((x+y)^{3}\)
- \((m+n)^{5}\)
- \((a+b)^{6}\)
- \((s+t)^{7}\)
- \((x-2)^{4}\)
- \((y-3)^{4}\)
- \((p-1)^{5}\)
- \((q-4)^{3}\)
- \((3x-y)^{5}\)
- \((5x-2y)^{4}\)
- \((2x+5y)^{4}\)
- \((3x+4y)^{5}\)
- Answer
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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. \(\begin{array}{l}{243 x^{5}+1620 x^{4} y+4320 x^{3} y^{2}} {+5760 x^{2} y^{3}+3840 x y^{4}+1024 y^{5}}\end{array}\)
Exercise \(\PageIndex{22}\) Use the Binomial Theorem to Expand a Binomial
In the following exercises, find the indicated term in the expansion of the binomial.
- Sixth term of \((x+y)^{10}\)
- Fifth term of \((a+b)^{9}\)
- Fourth term of \((x-y)^{8}\)
- Seventh term of \((x-y)^{11}\)
- Answer
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2. \(126a^{5} b^{4}\)
4. \(462x^{5} y^{6}\)
Exercise \(\PageIndex{23}\) Use the Binomial Theorem to Expand a Binomial
In the following exercises, find the coefficient of the indicated term in the expansion of the binomial.
- \(y^{3}\) term of \((y+5)^{4}\)
- \(x^{6}\) term of \((x+2)^{8}\)
- \(x^{5}\) term of \((x-4)^{6}\)
- \(x^{7}\) term of \((x-3)^{9}\)
- \(a^{4} b^{2}\) term of \((2 a+b)^{6}\)
- \(p^{5} q^{4}\) term of \((3 p+q)^{9}\)
- Answer
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2. \(112\)
4. \(324\)
6. \(30,618\)
Exercise \(\PageIndex{24}\) Writing Exercises
- In your own words explain how to find the rows of the Pascal's Triangle. Write the first five rows of Pascal's Triangle.
- In your own words, explain the pattern of exponents for each variable in the expansion of.
- In your own words, explain the difference between \((a+b)^{n}\) and \((a-b)^{n}\).
- In your own words, explain how to find a specific term in the expansion of a binomial without expanding the whole thing. Use an example to help explain.
- Answer
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2. Answers will vary
4. Answers will vary
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
b. On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?