# Using binomial theorem, write down the expansions of the following:

Question:

Using binomial theorem, write down the expansions of the following:

(i) $(2 x+3 y)^{5}$

(ii) $(2 x-3 y)^{4}$

(iii) $\left(x-\frac{1}{x}\right)^{6}$

(iv) $(1-3 x)^{7}$

(v) $\left(a x-\frac{b}{x}\right)^{6}$

(vi) $\left(\frac{\sqrt{x}}{a}-\sqrt{\frac{a}{x}}\right)^{6}$

(vii) $(\sqrt[3]{x}-\sqrt[3]{a})^{6}$

(viii) $\left(1+2 x-3 x^{2}\right)^{5}$

(ix) $\left(x+1-\frac{1}{x}\right)$

(x) $\left(1-2 x+3 x^{2}\right)^{3}$

Solution:

(i) $(2 x+3 y)^{5}$

$={ }^{5} C_{0}(2 x)^{5}(3 y)^{0}+{ }^{5} C_{1}(2 x)^{4}(3 y)^{1}+{ }^{5} C_{2}(2 x)^{3}(3 y)^{2}+{ }^{5} C_{3}(2 x)^{2}(3 y)^{3}+{ }^{5} C_{4}(2 x)^{1}(3 y)^{4}+{ }^{5} C_{5}(2 x)^{0}(3 y)^{5}$

$=32 x^{5}+5 \times 16 x^{4} \times 3 y+10 \times 8 x^{3} \times 9 y^{2}+10 \times 4 x^{2} \times 27 y^{3}+5 \times 2 x \times 81 y^{4}+243 y^{5}$

$=32 x^{5}+240 x^{4} y+720 x^{3} y^{2}+1080 x^{2} y^{3}+810 x y^{4}+243 y^{5}$

(ii) $(2 x-3 y)^{4}$

$={ }^{4} C_{0}(2 x)^{4}(3 y)^{0}-{ }^{4} C_{1}(2 x)^{3}(3 y)^{1}+{ }^{4} C_{2}(2 x)^{2}(3 y)^{2}-{ }^{4} C_{3}(2 x)^{1}(3 y)^{3}+{ }^{4} C_{4}(2 x)^{0}(3 y)^{4}$\

$=16 x^{4}-4 \times 8 x^{3} \times 3 y+6 \times 4 x^{2} \times 9 y^{2}-4 \times 2 x \times 27 y^{3}+81 y^{4}$

$=16 x^{4}-96 x^{3} y+216 x^{2} y^{2}-216 x y^{3}+81 y^{4}$

(iii) $\left(x-\frac{1}{x}\right)^{6}$

$={ }^{6} C_{0} x^{6}\left(\frac{1}{x}\right)^{0}-{ }^{6} C_{1} x^{5}\left(\frac{1}{x}\right)^{1}+{ }^{6} C_{2} x^{4}\left(\frac{1}{x}\right)^{2}-{ }^{6} C_{3} x^{3}\left(\frac{1}{x}\right)^{3}+{ }^{6} C_{4} x^{2}\left(\frac{1}{x}\right)^{4}-{ }^{6} C_{5} x^{1}\left(\frac{1}{x}\right)^{5}+{ }^{6} C_{6} x^{0}\left(\frac{1}{x}\right)^{6}$

$=x^{6}-6 x^{5} \times \frac{1}{x}+15 x^{4} \times \frac{1}{x^{2}}-20 x^{3} \times \frac{1}{x^{3}}+15 x^{2} \times \frac{1}{x^{4}}-6 x \times \frac{1}{x^{5}}+\frac{1}{x^{6}}$

$=x^{6}-6 x^{4}+15 x^{2}-20+\frac{15}{x^{2}}-\frac{6}{x^{4}}+\frac{1}{x^{6}}$

(iv) $(1-3 x)^{7}$

$={ }^{7} C_{0}(3 x)^{0}-{ }^{7} C_{1}(3 x)^{1}+{ }^{7} C_{2}(3 x)^{2}-{ }^{7} C_{3}(3 x)^{3}+{ }^{7} C_{4}(3 x)^{4}-{ }^{7} C_{5}(3 x)^{5}+{ }^{7} C_{6}(3 x)^{6}-{ }^{7} C_{7}(3 x)^{7}$

$=1-7 \times 3 x+21 \times 9 x^{2}-35 \times 27 x^{3}+35 \times 81 x^{4}-21 \times 243 x^{5}+7 \times 729 x^{6}-2187 x^{7}$

$=1-21 x+189 x^{2}-945 x^{3}+2835 x^{4}-5103 x^{5}+5103 x^{6}-2187 x^{7}$

(v) $\left(a x-\frac{b}{x}\right)^{6}$

$={ }^{6} C_{0}(a x)^{6}\left(\frac{b}{x}\right)^{0}-{ }^{6} C_{1}(a x)^{5}\left(\frac{b}{x}\right)^{1}+{ }^{6} C_{2}(a x)^{4}\left(\frac{b}{x}\right)^{2}-{ }^{6} C_{3}(a x)^{3}\left(\frac{b}{x}\right)^{3}+{ }^{6} C_{4}(a x)^{2}\left(\frac{b}{x}\right)^{4}-{ }^{6} C_{5}(a x)^{1}\left(\frac{b}{x}\right)^{5}+{ }^{6} C_{6}(a x)^{0}\left(\frac{b}{x}\right)^{6}$

$=a^{6} x^{6}-6 a^{5} x^{5} \times \frac{b}{x}+15 a^{4} x^{4} \times \frac{b^{2}}{x^{2}}-20 a^{3} b^{3} \times \frac{b^{3}}{x^{3}}+15 a^{2} x^{2} \times \frac{b^{4}}{x^{4}}-6 a x \times \frac{b^{5}}{x^{5}}+\frac{b^{6}}{x^{6}}$

$=a^{6} x^{6}-6 a^{5} x^{4} b+15 a^{4} x^{2} b^{2}-20 a^{3} b^{3}+15 \frac{a^{2} b^{4}}{x^{2}}-6 \frac{a b^{5}}{x^{4}}+\frac{b^{6}}{x^{6}}$

(vi) $\left(\frac{\sqrt{x}}{a}-\sqrt{\frac{a}{x}}\right)^{6}$

$={ }^{6} C_{0}\left(\sqrt{\frac{x}{a}}\right)^{6}\left(\sqrt{\frac{a}{x}}\right)^{0}-{ }^{6} C_{1}\left(\sqrt{\frac{x}{a}}\right)^{5}\left(\sqrt{\frac{a}{x}}\right)^{1}+{ }^{6} C_{2}\left(\sqrt{\frac{x}{a}}\right)^{4}\left(\sqrt{\frac{a}{x}}\right)^{2}-{ }^{6} C_{3}\left(\sqrt{\frac{x}{a}}\right)^{3}\left(\sqrt{\frac{a}{x}}\right)^{3}+{ }^{6} C_{4}\left(\sqrt{\frac{x}{a}}\right)^{2}\left(\sqrt{\frac{a}{x}}\right)^{4}$

$-{ }^{6} C_{5}\left(\sqrt{\frac{x}{a}}\right)^{1}\left(\sqrt{\frac{a}{x}}\right)^{5}+{ }^{6} C_{6}\left(\sqrt{\frac{x}{a}}\right)^{0}\left(\sqrt{\frac{a}{x}}\right)^{6}$

$=\frac{x^{3}}{a^{3}}-6 \frac{x^{2}}{a^{2}}+15 \frac{x}{a}-20+15 \frac{a}{x}-6 \frac{a^{2}}{x^{2}}+\frac{a^{3}}{x^{3}}$

(vii) $(\sqrt[3]{x}-\sqrt[3]{a})^{6}$

$={ }^{6} C_{0}(\sqrt[3]{x})^{6}(\sqrt[3]{a})^{0}-{ }^{6} C_{1}(\sqrt[3]{x})^{5}(\sqrt[3]{a})^{1}+{ }^{6} C_{2}(\sqrt[3]{x})^{4}(\sqrt[3]{a})^{2}-{ }^{6} C_{3}(\sqrt[3]{x})^{3}(\sqrt[3]{a})^{3}+{ }^{6} C_{4}(\sqrt[3]{x})^{2}(\sqrt[3]{a})^{4}$

$-{ }^{6} C_{5}(\sqrt[3]{x})^{1}(\sqrt[3]{a})^{5}+{ }^{6} C_{6}(\sqrt[3]{x})^{0}(\sqrt[3]{a})^{6}$

$=x^{2}-6 x^{5 / 3} a^{1 / 3}+15 x^{4 / 3} a^{2 / 3}-20 x a+15 x^{2 / 3} a^{4 / 3}-6 x^{1 / 3} a^{5 / 3}+a^{2}$

(viii) $\left(1+2 x-3 x^{2}\right)^{5}$

Consider $1-2 x$ and $3 x^{2}$ as two separate entities and apply the binomial theorem.

Now,

{ }^{5} C_{0}(1+2 x)^{5}(3 x)^{0}-{ }^{5} C_{1}(1+2 x)^{4}\left(3 x^{2}\right)^{1}+{ }^{5} C_{2}(1+2 x)^{3}\left(3 x^{2}\right)^{2}-{ }^{5} C_{3}(1+2 x)^{2}\left(3 x^{2}\right)^{3}+{ }^{5} C_{4}(1+2 x)^{1}\left(3 x^{2}\right)^{4}-{ }^{5} C_{5}(1+2 x)^{0}\left(3 x^{2}\right)^{5} \begin{aligned}=&{ }^{5} C_{0} \times(2 x)^{0}+{ }^{5} C_{1} \times(2 x)^{1}+{ }^{5} C_{2} \times(2 x)^{2}+{ }^{5} C_{3} \times(2 x)^{3}+{ }^{5} C_{4} \times(2 x)^{4}+{ }^{5} C_{5} \times(2 x)^{5}-\\ & 15 x^{2}\left[{ }^{4} C_{0}(2 x)^{0}+{ }^{4} C_{1}(2 x)^{1}+{ }^{4} C_{2}(2 x)^{2}+{ }^{4} C_{3}(2 x)^{3}+{ }^{4} C_{4}(2 x)^{4}\right]+\\ & 90 x^{4}\left[1+8 x^{3}+6 x+12 x^{2}\right]-270 x^{6}\left(1+4 x^{2}+4 x\right)+405 x^{8}+810 x^{9}-243 x^{10} \end{aligned} =1+10 x+40 x^{2}+80 x^{3}+80 x^{4}+32 x^{5}-15 x^{2}-120 x^{3}-360^{4}-480 x^{5}-240 x^{6}+90 x^{4}+720 x^{7}+540 x^{5}+1080 x^{6}-270 x^{6}-1080 x^{8}-1080 x^{7}+405 x^{8}+810 x^{9}-243 x^{10}

$=1+10 x+25 x^{2}-40 x^{3}-190 x^{4}+92 x^{5}+570 x^{6}-360 x^{7}-675 x^{8}+810 x^{9}-243 x^{10}$

(ix) $\left(x+1-\frac{1}{x}\right)^{3}$

$={ }^{3} C_{0}(x+1)^{3}\left(\frac{1}{x}\right)^{0}-{ }^{3} C_{1}(x+1)^{2}\left(\frac{1}{x}\right)^{1}+{ }^{3} C_{2}(x+1)^{1}\left(\frac{1}{x}\right)^{2}-{ }^{3} C_{3}(x+1)^{0}\left(\frac{1}{x}\right)^{3}$

$=(x+1)^{3}-3(x+1)^{2} \times \frac{1}{x}+3 \frac{x+1}{x^{2}}-\frac{1}{x^{3}}$

$=x^{3}+1+3 x+3 x^{2}-\frac{3 x^{2}+3+6 x}{x}+3 \frac{x+1}{x^{2}}-\frac{1}{x^{3}}$

$=x^{3}+1+3 x+3 x^{2}-3 x-\frac{3}{x}-6+\frac{3}{x}+\frac{3}{x^{2}}-\frac{1}{x^{3}}$

$=x^{3}+3 x^{2}-5+\frac{3}{x^{2}}-\frac{1}{x^{3}}$

(x) $\left(1-2 x+3 x^{2}\right)^{3}$

$={ }^{3} C_{0}(1-2 x)^{3}+{ }^{3} C_{1}(1-2 x)^{2}\left(3 x^{2}\right)+{ }^{3} C_{2}(1-2 x)\left(3 x^{2}\right)^{2}+{ }^{3} C_{3}\left(3 x^{2}\right)^{3}$

$=(1-2 x)^{3}+9 x^{2}(1-2 x)^{2}+27 x^{4}(1-2 x)+27 x^{6}$

$=1-8 x^{3}+12 x^{2}-6 x+9 x^{2}\left(1+4 x^{2}-4 x\right)+27 x^{4}-54 x^{5}+27 x^{6}$

$=1-8 x^{3}+12 x^{2}-6 x+9 x^{2}+36 x^{4}-36 x^{3}+27 x^{4}-54 x^{5}+27 x^{6}$

$=1-6 x+21 x^{2}-44 x^{3}+63 x^{4}-54 x^{5}+27 x^{6}$