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Study Guides > College Algebra

Solutions

Solutions to Try Its

1. a. 35 b. 330 2. a. x55x4y+10x3y210x2y3+5xy4y5{x}^{5}-5{x}^{4}y+10{x}^{3}{y}^{2}-10{x}^{2}{y}^{3}+5x{y}^{4}-{y}^{5} b. 8x3+60x2y+150xy2+125y38{x}^{3}+60{x}^{2}y+150x{y}^{2}+125{y}^{3} 3. 10,206x4y5-10,206{x}^{4}{y}^{5}

Solutions to Odd-Numbered Exercises

1. A binomial coefficient is an alternative way of denoting the combination C(n,r)C\left(n,r\right). It is defined as (nr)=C(n,r)=n!r!(nr)!\left(\begin{array}{c}n\\ r\end{array}\right)=C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}. 3. The Binomial Theorem is defined as (x+y)n=k=0n(nk)xnkyk{\left(x+y\right)}^{n}=\sum _{k=0}^{n}\left(\begin{array}{c}n\\ k\end{array}\right){x}^{n-k}{y}^{k} and can be used to expand any binomial. 5. 15 7. 35 9. 10 11. 12,376 13. 64a348a2b+12ab2b364{a}^{3}-48{a}^{2}b+12a{b}^{2}-{b}^{3} 15. 27a3+54a2b+36ab2+8b327{a}^{3}+54{a}^{2}b+36a{b}^{2}+8{b}^{3} 17. 1024x5+2560x4y+2560x3y2+1280x2y3+320xy4+32y51024{x}^{5}+2560{x}^{4}y+2560{x}^{3}{y}^{2}+1280{x}^{2}{y}^{3}+320x{y}^{4}+32{y}^{5} 19. 1024x53840x4y+5760x3y24320x2y3+1620xy4243y51024{x}^{5}-3840{x}^{4}y+5760{x}^{3}{y}^{2}-4320{x}^{2}{y}^{3}+1620x{y}^{4}-243{y}^{5} 21. 1x4+8x3y+24x2y2+32xy3+16y4\frac{1}{{x}^{4}}+\frac{8}{{x}^{3}y}+\frac{24}{{x}^{2}{y}^{2}}+\frac{32}{x{y}^{3}}+\frac{16}{{y}^{4}} 23. a17+17a16b+136a15b2{a}^{17}+17{a}^{16}b+136{a}^{15}{b}^{2} 25. a1530a14b+420a13b2{a}^{15}-30{a}^{14}b+420{a}^{13}{b}^{2} 27. 3,486,784,401a20+23,245,229,340a19b+73,609,892,910a18b23,486,784,401{a}^{20}+23,245,229,340{a}^{19}b+73,609,892,910{a}^{18}{b}^{2} 29. x248x21y+28x18y{x}^{24}-8{x}^{21}\sqrt{y}+28{x}^{18}y 31. 720x2y3-720{x}^{2}{y}^{3} 33. 220,812,466,875,000y7220,812,466,875,000{y}^{7} 35. 35x3y435{x}^{3}{y}^{4} 37. 1,082,565a3b161,082,565{a}^{3}{b}^{16} 39. 1152y2x7\frac{1152{y}^{2}}{{x}^{7}} 41. f2(x)=x4+12x3{f}_{2}\left(x\right)={x}^{4}+12{x}^{3} Graph of the function f_2. 43. f4(x)=x4+12x3+54x2+108x{f}_{4}\left(x\right)={x}^{4}+12{x}^{3}+54{x}^{2}+108x Graph of the function f_4. 45. 590,625x5y2590,625{x}^{5}{y}^{2} 47. (nk1)+(nk)=(n+1k)\left(\begin{array}{c}n\\ k - 1\end{array}\right)+\left(\begin{array}{l}n\\ k\end{array}\right)=\left(\begin{array}{c}n+1\\ k\end{array}\right); Proof: (nk1)+(nk)=n!k!(nk)!+n!(k1)!(n(k1))!=n!k!(nk)!+n!(k1)!(nk+1)!=(nk+1)n!(nk+1)k!(nk)!+kn!k(k1)!(nk+1)!=(nk+1)n!+kn!k!(nk+1)!=(n+1)n!k!((n+1)k)!=(n+1)!k!((n+1)k)!=(n+1k)\begin{array}{}\\ \\ \\ \left(\begin{array}{c}n\\ k - 1\end{array}\right)+\left(\begin{array}{l}n\\ k\end{array}\right)\\ =\frac{n!}{k!\left(n-k\right)!}+\frac{n!}{\left(k - 1\right)!\left(n-\left(k - 1\right)\right)!}\\ =\frac{n!}{k!\left(n-k\right)!}+\frac{n!}{\left(k - 1\right)!\left(n-k+1\right)!}\\ =\frac{\left(n-k+1\right)n!}{\left(n-k+1\right)k!\left(n-k\right)!}+\frac{kn!}{k\left(k - 1\right)!\left(n-k+1\right)!}\\ =\frac{\left(n-k+1\right)n!+kn!}{k!\left(n-k+1\right)!}\\ =\frac{\left(n+1\right)n!}{k!\left(\left(n+1\right)-k\right)!}\\ =\frac{\left(n+1\right)!}{k!\left(\left(n+1\right)-k\right)!}\\ =\left(\begin{array}{c}n+1\\ k\end{array}\right)\end{array} 49. The expression (x3+2y2z)5{\left({x}^{3}+2{y}^{2}-z\right)}^{5} cannot be expanded using the Binomial Theorem because it cannot be rewritten as a binomial.

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