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Greatest Binomial Coefficients

Using binomia...

Question

Using binomial theorem, write down the expansions of the following:
$(1 + 2 x - 3 x^{2})^{5}$

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Solution

$(1 + 2 x - 3 x^{2})^{5}$

$= [(1 + 3 x) (1 - x)]^{5}$

$= (1 + 3 x)^{5} (1 - x)^{5}$

$= [1 + 5 (3 x) + 10 (3 x)^{2} + 10 (3 x)^{3} + 5 (3 x)^{4} + (3 x)^{5}] [1 - 5 x + 10 x^{2} - 10 x^{3} + 5 x^{4} - x^{5}]$

$= [1 + 15 x + 90 x^{2} + 270 x^{3} + 405 x^{4} + 243 x^{5}] [1 - 5 x + 10 x^{2} - 10 x^{3} + 5 x^{4} - x^{5}]$

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Similar questions

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

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

$(i i i) {(x - \frac{1}{x})}^{6}$

$(i v) (1 - 3 x)^{7}$

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

$(v i) {(\sqrt{\frac{x}{a}} - \sqrt{\frac{a}{x}})}^{6}$

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

$(v i i i) (1 + 2 x - 3 x^{2})^{5}$

$(i x) {(x + 1 - \frac{1}{x})}^{3}$

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

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

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

{(x - \frac{1}{x})}^{6}

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

{(a x - \frac{b}{x})}^{6}

{(\frac{\sqrt{x}}{a} - \sqrt{\frac{a}{x}})}^{6}

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

{(1 + 2 x - 3 x^{2})}^{5}

(x + 1 - \frac{1}{x})

{(1 - 2 x + 3 x^{2})}^{3}

(4 p + 5 q)^{8}

Find the expansion of using binomial theorem.

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