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

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

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