![\"Graph](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25202620\/CNX_Precalc_Figure_11_06_2022.jpg\")
\n\n43.\u00a0[latex]{f}_{4}\\left(x\\right)={x}^{4}+12{x}^{3}+54{x}^{2}+108x[\/latex]\n![\"Graph](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25202621\/CNX_Precalc_Figure_11_06_2042.jpg\")
\n\n45.\u00a0[latex]590,625{x}^{5}{y}^{2}[\/latex]\n\n47.\u00a0[latex]\\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)[\/latex]; Proof:\n[latex]\\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}[\/latex]\n\n49.\u00a0The expression [latex]{\\left({x}^{3}+2{y}^{2}-z\\right)}^{5}[\/latex] cannot be expanded using the Binomial Theorem because it cannot be rewritten as a binomial.","rendered":"1. a.\u00a035
\nb.\u00a0330<\/p>\n
2. a.\u00a0[latex]{x}^{5}-5{x}^{4}y+10{x}^{3}{y}^{2}-10{x}^{2}{y}^{3}+5x{y}^{4}-{y}^{5}[\/latex]
\nb.\u00a0[latex]8{x}^{3}+60{x}^{2}y+150x{y}^{2}+125{y}^{3}[\/latex]<\/p>\n
3.\u00a0[latex]-10,206{x}^{4}{y}^{5}[\/latex]<\/p>\n
1.\u00a0A binomial coefficient is an alternative way of denoting the combination [latex]C\\left(n,r\\right)[\/latex]. It is defined as [latex]\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right)=C\\left(n,r\\right)=\\frac{n!}{r!\\left(n-r\\right)!}[\/latex].<\/p>\n
3.\u00a0The Binomial Theorem is defined as [latex]{\\left(x+y\\right)}^{n}=\\sum _{k=0}^{n}\\left(\\begin{array}{c}n\\\\ k\\end{array}\\right){x}^{n-k}{y}^{k}[\/latex] and can be used to expand any binomial.<\/p>\n
5.\u00a015<\/p>\n
7.\u00a035<\/p>\n
9.\u00a010<\/p>\n
11.\u00a012,376<\/p>\n
13.\u00a0[latex]64{a}^{3}-48{a}^{2}b+12a{b}^{2}-{b}^{3}[\/latex]<\/p>\n
15.\u00a0[latex]27{a}^{3}+54{a}^{2}b+36a{b}^{2}+8{b}^{3}[\/latex]<\/p>\n
17.\u00a0[latex]1024{x}^{5}+2560{x}^{4}y+2560{x}^{3}{y}^{2}+1280{x}^{2}{y}^{3}+320x{y}^{4}+32{y}^{5}[\/latex]<\/p>\n
19.\u00a0[latex]1024{x}^{5}-3840{x}^{4}y+5760{x}^{3}{y}^{2}-4320{x}^{2}{y}^{3}+1620x{y}^{4}-243{y}^{5}[\/latex]<\/p>\n
21.\u00a0[latex]\\frac{1}{{x}^{4}}+\\frac{8}{{x}^{3}y}+\\frac{24}{{x}^{2}{y}^{2}}+\\frac{32}{x{y}^{3}}+\\frac{16}{{y}^{4}}[\/latex]<\/p>\n
23.\u00a0[latex]{a}^{17}+17{a}^{16}b+136{a}^{15}{b}^{2}[\/latex]<\/p>\n
25.\u00a0[latex]{a}^{15}-30{a}^{14}b+420{a}^{13}{b}^{2}[\/latex]<\/p>\n
27.\u00a0[latex]3,486,784,401{a}^{20}+23,245,229,340{a}^{19}b+73,609,892,910{a}^{18}{b}^{2}[\/latex]<\/p>\n
29.\u00a0[latex]{x}^{24}-8{x}^{21}\\sqrt{y}+28{x}^{18}y[\/latex]<\/p>\n
31.\u00a0[latex]-720{x}^{2}{y}^{3}[\/latex]<\/p>\n
33.\u00a0[latex]220,812,466,875,000{y}^{7}[\/latex]<\/p>\n
35.\u00a0[latex]35{x}^{3}{y}^{4}[\/latex]<\/p>\n
37.\u00a0[latex]1,082,565{a}^{3}{b}^{16}[\/latex]<\/p>\n
39.\u00a0[latex]\\frac{1152{y}^{2}}{{x}^{7}}[\/latex]<\/p>\n
41.\u00a0[latex]{f}_{2}\\left(x\\right)={x}^{4}+12{x}^{3}[\/latex]
\n<\/p>\n
43.\u00a0[latex]{f}_{4}\\left(x\\right)={x}^{4}+12{x}^{3}+54{x}^{2}+108x[\/latex]
\n<\/p>\n
45.\u00a0[latex]590,625{x}^{5}{y}^{2}[\/latex]<\/p>\n
47.\u00a0[latex]\\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)[\/latex]; Proof:
\n[latex]\\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}[\/latex]<\/p>\n
49.\u00a0The expression [latex]{\\left({x}^{3}+2{y}^{2}-z\\right)}^{5}[\/latex] cannot be expanded using the Binomial Theorem because it cannot be rewritten as a binomial.<\/p>\n\n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t
\n\t\t\t\t\t\t \n\t\t\t\t\t\t\t CC licensed content, Specific attribution<\/div>