Solving Lesson 4 Page 37 Math Learning Topic 10 – Kite>

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Topic

Determine the coefficient of:

a) \({x^{12}}\) in the expansion of the expression \({(x + 4)^{30}}\)

b) \({x^{10}}\) in the expansion of the expression \({(3 + 2x)^{30}}\)

c) \({x^{15}}\) and \({x^{16}}\) in the expansion of the expression \({\left( {\frac{{2x}}{3} – \) frac{1}{7}} \right)^{51}}\)

Solution method – See details

Newton’s binomial formula: \({(a + b)^n} = C_n^0{a^n} + C_n^1{a^{n – 1}}b + … + C_n^{n – 1} a{b^{n – 1}} + C_n^n{b^n}\)

Detailed explanation

a) According to Newton’s binomial formula, we have:

\({(x + 4)^{30}} = C_{30}^0{x^{30}} + C_{30}^1{x^{29}}{4^1} + … + C_ {30}^k{x^{30 – k}}{4^k} + … + C_{30}^{30}{4^{30}}\)

The term containing \({x^{12}}\) corresponds to \(30 – k = 12 \Rightarrow k = 18\). Therefore the coefficient of \({x^{12}}\) is

\(C_{30}^{18}{4^{18}}\)

b) According to Newton’s binomial formula, we have:

\({(3 + 2x)^{30}} = C_{30}^0{3^{30}} + C_{30}^1{3^{29}}{\left( {2x} \right )^1} + … + C_{30}^k{3^{30 – k}}{\left( {2x} \right)^k} + … + C_{30}^{30}{\left( {2x} \right)^{30}}\)

The term containing \({x^{10}}\) corresponds to \(k = 10\). Therefore the coefficient of \({x^{10}}\) is

\(C_{30}^{10}{3^{20}}{2^{10}}\)

c) According to Newton’s binomial formula, we have:

\({\left( {\frac{{2x}}{3} – \frac{1}{7}} \right)^{51}} = C_{51}^0{\left( {\frac{ {2x}}{3}} \right)^{51}} + C_{51}^1{\left( {\frac{{2x}}{3}} \right)^{50}}{\left ( { – \frac{1}{7}} \right)^1} + … + C_{51}^k{\left( {\frac{{2x}}{3}} \right)^{51 – k}}{\left( { – \frac{1}{7}} \right)^k} + … + C_{51}^{51}{\left( { – \frac{1}{7}} \right)^{51}}\)

The term containing \({x^{15}}\) corresponds to \(51 – k = 15 \Leftrightarrow k = 36\). So the coefficient of \({x^{15}}\) is

\(C_{51}^{15}{\left( {\frac{2}{3}} \right)^{15}}{\left( { – \frac{1}{7}} \right) ^{36}}\)

The term containing \({x^{16}}\) corresponds to \(51 – k = 16 \Leftrightarrow k = 35\). Therefore the coefficient of \({x^{16}}\) is

\(C_{51}^{16}{\left( {\frac{2}{3}} \right)^{16}}{\left( { – \frac{1}{7}} \right) ^{35}}\)

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