# 12.5E: 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.

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. $$(3 x+5 y)^{3}$$

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

2.

1. $$7$$
2. $$1$$
3. $$1$$
4. $$45$$

4.

1. $$4$$
2. $$1$$
3. $$1$$
4. $$55$$
##### Exercise $$\PageIndex{21}$$ Use the Binomial Theorem to Expand a Binomial

In the following exercises, expand each 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. $$(3x+4y)^{5}$$

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.

1. Sixth term of $$(x+y)^{10}$$
2. Fifth term of $$(a+b)^{9}$$
3. Fourth term of $$(x-y)^{8}$$
4. Seventh term of $$(x-y)^{11}$$

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.

1. $$y^{3}$$ term of $$(y+5)^{4}$$
2. $$x^{6}$$ term of $$(x+2)^{8}$$
3. $$x^{5}$$ term of $$(x-4)^{6}$$
4. $$x^{7}$$ term of $$(x-3)^{9}$$
5. $$a^{4} b^{2}$$ term of $$(2 a+b)^{6}$$
6. $$p^{5} q^{4}$$ term of $$(3 p+q)^{9}$$

2. $$112$$

4. $$324$$

6. $$30,618$$

##### Exercise $$\PageIndex{24}$$ Writing Exercises
1. In your own words explain how to find the rows of the Pascal's Triangle. Write the first five rows of Pascal's Triangle.
2. In your own words, explain the pattern of exponents for each variable in the expansion of.
3. In your own words, explain the difference between $$(a+b)^{n}$$ and $$(a-b)^{n}$$.
4. 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.