Question

Using binomial theorem, write down the expansions of the following:
(i) ${\left(2x+3y\right)}^5$

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

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

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

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

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

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

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

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

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

Answer 1

(i) (2x + 3y)⁵

$={}^{5}C_0\left(2x{)}^5\right(3y{)}^0+{}^{5}C_1\left(2x{)}^4\right(3y{)}^1+{}^{5}C_2\left(2x{)}^3\right(3y{)}^2+{}^{5}C_3\left(2x{)}^2\right(3y{)}^3+{}^{5}C_4\left(2x{)}^1\right(3y{)}^4+{}^{5}C_5\left(2x{)}^0\right(3y{)}^5$
$$\begin{gathered} =32x^5+5\times 16x^4\times 3y+10\times 8x^3\times 9y^2+10\times 4x^2\times 27y^3+5\times 2x\times 81y^4+243y^5 \\ =32x^5+240x^{4}y+720x^3{y}^2+1080x^2{y}^3+810x{y}^4+243y^5 \end{gathered}$$

(ii) (2x − 3y)<sup>4

$$\begin{gathered} ={}^{4}C_0\left(2x{)}^4\right(3y{)}^0-{}^{4}C_1\left(2x{)}^3\right(3y{)}^1+{}^{4}C_2\left(2x{)}^2\right(3y{)}^2-{}^{4}C_3\left(2x{)}^1\right(3y{)}^3+{}^{4}C_4\left(2x{)}^0\right(3y{)}^4 \\ =16x^4-4\times 8x^3\times 3y+6\times 4x^2\times 9y^2-4\times 2x\times 27y^3+81y^4 \\ =16x^4-96x^{3}y+216x^2{y}^2-216x{y}^3+81y^4 \end{gathered}$$</sup>

(iii)
$$\begin{gathered} {\left(x-\frac{1}{x}\right)}^6 \\ ={}^{6}C_0{x}^6{\left(\frac{1}{x}\right)}^0-{}^{6}C_1{x}^5{\left(\frac{1}{x}\right)}^1+{}^{6}C_2{x}^4{\left(\frac{1}{x}\right)}^2-{}^{6}C_3{x}^3{\left(\frac{1}{x}\right)}^3+{}^{6}C_4{x}^2{\left(\frac{1}{x}\right)}^4{-}^6{C}_5{x}^1{\left(\frac{1}{x}\right)}^5+{}^{6}C_6{x}^0{\left(\frac{1}{x}\right)}^6 \\ =x^6-6x^5\times \frac{1}{x}+15x^4\times \frac{1}{x^2}-20x^3\times \frac{1}{x^3}+15x^2\times \frac{1}{x^4}-6x\times \frac{1}{x^5}+\frac{1}{x^6} \\ =x^6-6x^4+15x^2-20+\frac{15}{x^2}-\frac{6}{x^4}+\frac{1}{x^6} \end{gathered}$$

(iv) (1 − 3x)<sup>7
$$\begin{gathered} ={}^{7}C_0(3x{)}^0-{}^{7}C_1(3x{)}^1+{}^{7}C_2(3x{)}^2-{}^{7}C_3(3x{)}^3+{}^{7}C_4(3x{)}^4-{}^{7}C_5(3x{)}^5+{}^{7}C_6(3x{)}^6-{}^{7}C_7(3x{)}^7 \\ =1-7\times 3x+21\times 9x^2-35\times 27x^3+35\times 81x^4-21\times 243x^5+7\times 729x^6-2187x^7 \\ =1-21x+189x^2-945x^3+2835x^4-5103x^5+5103x^6-2187x^7 \end{gathered}$$

(v)</sup>
$$\begin{gathered} (ax-\frac{b}{x}{)}^6 \\ ={}^{6}C_0(ax{)}^6(\frac{b}{x}{)}^0-{}^{6}C_1(ax{)}^5(\frac{b}{x}{)}^1+{}^{6}C_2(ax{)}^4(\frac{b}{x}{)}^2-{}^{6}C_3(ax{)}^3(\frac{b}{x}{)}^3+{}^{6}C_4(ax{)}^2(\frac{b}{x}{)}^4-{}^{6}C_5(ax{)}^1(\frac{b}{x}{)}^5+{}^{6}C_6(ax{)}^0(\frac{b}{x}{)}^6 \end{gathered}$$
$$\begin{gathered} =a^6{x}^6-6a^5{x}^5\times \frac{b}{x}+15a^4{x}^4\times \frac{b^2}{x^2}-20a^3{b}^3\times \frac{b^3}{x^3}+15a^2{x}^2\times \frac{b^4}{x^4}-6ax\times \frac{b^5}{x^5}+\frac{b^6}{x^6} \\ =a^6{x}^6-6a^5{x}^{4}b+15a^4{x}^2{b}^2-20a^3{b}^3+15\frac{a^2{b}^4}{x^2}-6\frac{a{b}^5}{x^4}+\frac{b^6}{x^6} \end{gathered}$$

(vi)
$$\begin{gathered} {\left(\sqrt{\frac{x}{a}}-\sqrt{\frac{a}{x}}\right)}^6 \\ ={}^{6}C_0{\left(\sqrt{\frac{x}{a}}\right)}^6{\left(\sqrt{\frac{a}{x}}\right)}^0-{}^{6}C_1{\left(\sqrt{\frac{x}{a}}\right)}^5{\left(\sqrt{\frac{a}{x}}\right)}^1+{}^{6}C_2{\left(\sqrt{\frac{x}{a}}\right)}^4{\left(\sqrt{\frac{a}{x}}\right)}^2-{}^{6}C_3{\left(\sqrt{\frac{x}{a}}\right)}^3{\left(\sqrt{\frac{a}{x}}\right)}^3+{}^{6}C_4{\left(\sqrt{\frac{x}{a}}\right)}^2{\left(\sqrt{\frac{a}{x}}\right)}^4-{}^{6}C_5{\left(\sqrt{\frac{x}{a}}\right)}^1{\left(\sqrt{\frac{a}{x}}\right)}^5+{}^{6}C_6{\left(\sqrt{\frac{x}{a}}\right)}^0{\left(\sqrt{\frac{a}{x}}\right)}^6 \\ =\frac{x^3}{a^3}-6\frac{x^2}{a^2}+15\frac{x}{a}-20+15\frac{a}{x}-6\frac{a^2}{x^2}+\frac{a^3}{x^3} \end{gathered}$$

(vii)
$$\begin{gathered} {\left(\sqrt[3]{x}-\sqrt[3]{a}\right)}^6 \\ ={}^{6}C_0\left(\sqrt[3]{x}{)}^6\right(\sqrt[3]{a}{)}^0-{}^{6}C_1\left(\sqrt[3]{x}{)}^5\right(\sqrt[3]{a}{)}^1+{}^{6}C_2\left(\sqrt[3]{x}{)}^4\right(\sqrt[3]{a}{)}^2-{}^{6}C_3\left(\sqrt[3]{x}{)}^3\right(\sqrt[3]{a}{)}^3+{}^{6}C_4\left(\sqrt[3]{x}{)}^2\right(\sqrt[3]{a}{)}^4-{}^{6}C_5\left(\sqrt[3]{x}{)}^1\right(\sqrt[3]{a}{)}^5+{}^{6}C_6\left(\sqrt[3]{x}{)}^0\right(\sqrt[3]{a}{)}^6 \\ =x^2-6x^{5/3}{a}^{1/3}+15x^{4/3}{a}^{2/3}-20xa+15x^{2/3}{a}^{4/3}-6x^{1/3}{a}^{5/3}+a^2 \end{gathered}$$


(viii)
$$\begin{gathered} (1+2x-3x^2{)}^5 \\ \mathrm{Consider}1-2x\mathrm{and}3x^2\mathrm{as}\mathrm{two}\mathrm{separate}\mathrm{entities}\mathrm{and}\mathrm{apply}\mathrm{the}\mathrm{binomial}\mathrm{theorem}. \\ \mathrm{Now}, \\ {}^{5}C_0(1+2x{)}^5(3x{)}^0-{}^{5}C_1(1+2x{)}^4(3x^2{)}^1+{}^{5}C_2(1+2x{)}^3(3x^2{)}^2-{}^{5}C_3(1+2x{)}^2(3x^2{)}^3+{}^{5}C_4(1+2x{)}^1(3x^2{)}^4-{}^{5}C_5(1+2x{)}^0(3x^2{)}^5 \\ =(1+2x{)}^5-5(1+2x{)}^4\times 3x^2+10\times (1+2x{)}^3\times 9x^4-10\times (1+2x{)}^2\times 27x^6+5(1+2x)\times 81x^8-243x^{10} \\ ={}^{5}C_0\times (2x{)}^0+{}^{5}C_1\times (2x{)}^1+{}^{5}C_2\times (2x{)}^2+{}^{5}C_3\times (2x{)}^3+{}^{5}C_4\times (2x{)}^4+{}^{5}C_5\times (2x{)}^5- \\ 15x^2[{}^{4}C_0(2x{)}^0+{}^{4}C_1(2x{)}^1+{}^{4}C_2(2x{)}^2+{}^{4}C_3(2x{)}^3+{}^{4}C_4\left(2x{)}^4\right]+ \\ 90x^4[1+8x^3+6x+12x^2]-270x^6(1+4x^2+4x)+405x^8+810x^9-243x^{10} \\ =1+10x+40x^2+80x^3+80x^4+32x^5-15x^2-120x^3-{360}^4-480x^5-240x^6+ \\ 90x^4+720x^7+540x^5+1080x^6-270x^6-1080x^8-1080x^7+405x^8+810x^9-243x^{10} \\ =1+10x+25x^2-40x^3-190x^4+92x^5+570x^6-360x^7-675x^8+810x^9-243x^{10} \end{gathered}$$


(ix)
$$\begin{gathered} (x+1-\frac{1}{x}{)}^3 \\ ={}^{3}C_0(x+1{)}^3(\frac{1}{x}{)}^0-{}^{3}C_1(x+1{)}^2(\frac{1}{x}{)}^1+{}^{3}C_2(x+1{)}^1(\frac{1}{x}{)}^2-{}^{3}C_3(x+1{)}^0(\frac{1}{x}{)}^3 \end{gathered}$$
$$\begin{gathered} =(x+1{)}^3-3(x+1{)}^2\times \frac{1}{x}+3\frac{x+1}{x^2}-\frac{1}{x^3} \\ =x^3+1+3x+3x^2-\frac{3x^2+3+6x}{x}+3\frac{x+1}{x^2}-\frac{1}{x^3} \\ =x^3+1+3x+3x^2-3x-\frac{3}{x}-6+\frac{3}{x}+\frac{3}{x^2}-\frac{1}{x^3} \\ =x^3+3x^2-5+\frac{3}{x^2}-\frac{1}{x^3} \end{gathered}$$

(x)
$$\begin{gathered} (1-2x+3x^2{)}^3 \\ ={}^{3}C_0(1-2x{)}^3+{}^{3}C_1(1-2x{)}^2(3x^2)+{}^{3}C_2(1-2x\left)\right(3x^2{)}^2+{}^{3}C_3(3x^2{)}^3 \\ =(1-2x{)}^3+9x^2(1-2x{)}^2+27x^4(1-2x)+27x^6 \\ =1-8x^3+12x^2-6x+9x^2(1+4x^2-4x)+27x^4-54x^5+27x^6 \\ =1-8x^3+12x^2-6x+9x^2+36x^4-36x^3+27x^4-54x^5+27x^6 \\ =1-6x+21x^2-44x^3+63x^4-54x^5+27x^6 \end{gathered}$$