14.5E: Exercises

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Aug 24, 2020
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- 14.5: Binomial Theorem
- 14.6: Chapter 12 Review 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

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

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. $\left( \begin{array}{l}{11} \\ {9}\end{array}\right)$

Answer

$7$
$1$
$1$
$45$

$4$
$1$
$1$
$55$

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

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

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

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

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.

This figure shows a table with four rows and four columns. The first row is the header row and reads. “I can”, “Confidently”, “With some help” and “No, I don’t get it”. The first column, beginning at the second row reads, “Use Pascal’s Triangle to Expand a Binomial”, “Evaluate a Binomial Coefficient” and “Use the Binomial Theorem to Expand a Binomial”. The remaining columns are blank. — Figure 12.4.31

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?

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Practice Makes Perfect

Exercise $\PageIndex{19}$ Use Pascal's Triangle to Expand a Binomial

Exercise $\PageIndex{20}$ Evaluate a Binomial Coefficient

Exercise $\PageIndex{21}$ Use the Binomial Theorem to Expand a Binomial

Exercise $\PageIndex{22}$ Use the Binomial Theorem to Expand a Binomial

Exercise $\PageIndex{23}$ Use the Binomial Theorem to Expand a Binomial

Exercise $\PageIndex{24}$ Writing Exercises

Self Check

Practice Makes Perfect

Exercise 14.5E.19\PageIndex{19} Use Pascal's Triangle to Expand a Binomial

Exercise 14.5E.20\PageIndex{20} Evaluate a Binomial Coefficient

Exercise 14.5E.21\PageIndex{21} Use the Binomial Theorem to Expand a Binomial

Exercise 14.5E.22\PageIndex{22} Use the Binomial Theorem to Expand a Binomial

Exercise 14.5E.23\PageIndex{23} Use the Binomial Theorem to Expand a Binomial

Exercise 14.5E.24\PageIndex{24} Writing Exercises

Self Check

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Exercise $\PageIndex{19}$ Use Pascal's Triangle to Expand a Binomial

Exercise $\PageIndex{20}$ Evaluate a Binomial Coefficient

Exercise $\PageIndex{21}$ Use the Binomial Theorem to Expand a Binomial

Exercise $\PageIndex{22}$ Use the Binomial Theorem to Expand a Binomial

Exercise $\PageIndex{23}$ Use the Binomial Theorem to Expand a Binomial

Exercise $\PageIndex{24}$ Writing Exercises