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Binomial Expression

Find the coef...

Question

Find the coefficient of $x^{4}$ in the product $(1 + 2 x)^{4} \times (2 - x)^{5}$

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Solution

Using the binomial expansion, we get
$(1 + 2 x)^{4} =^{4} C_{0} +^{4} C_{0} +^{4} C_{1} (2 x) +^{4} C_{2} (2 x)^{2} +^{4} C_{3} (2 x)^{3} + 4 C_{4} (2 x)^{4} = 1 + 8 x + 24 x^{2} + 32 x^{3} + 16 x^{4} and (2 - x)^{5} = 2^{5} -^{5} C_{1} (4^{4}) x +^{5} C_{2} (2^{3}) x^{2} -^{5} C_{3} (2^{2}) x^{3} +^{5} C_{4} (2) x^{4} (2) x^{4} -^{5} C_{5} x^{5} = 32 - 80 x + 80 x^{2} - 40 x^{3} + 10 x^{4} - x^{5} ∴ (1 + 2 x)^{4} \times (2 - x)^{5} = (1 + 8 x + 24 x^{2} + 32 x^{3} + 16 x^{4}) \times (32 - 80 x + 80 x^{2} - 40 x^{3} + 10 x^{4} - x^{5}) sum of the terms containing x^{4} in the given product$
$= (1 \times 10 x^{4}) + 8 x (- 40 x^{3}) + (24 x^{2}) (80 x^{2}) + (32 x^{3}) (- 80 x) + (16 x^{4} \times 32) = 10 x^{4} - 320 x^{4} + 1920 x^{4} - 2560 x^{4} + 512 x^{2} = (10 - 320 + 1920 - 2560 + 512) x^{4} = - 438 x^{4} .$
Hence the coefficient of $x^{4}$ in the given product is -438

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Similar questions

Q. Find the coefficient of

x^{4}

in the expansion of the product

(1 + 2 x)^{4} (2 - x)^{5}

Q. In the product of expression of

(1 + 2 x)^{6} (1 - x)^{7}

, find the coefficient of

x^{5}

Q. Find the coefficient of

x^{5}

in the expansion of

(1 + x^{2})^{5} (1 + x)^{4}

Find the polynomial from the given information.

Quotient = x^{3} + 2 x

Divisor = x^{2} + \frac{1}{2} x

Remainder = 5

Q. Coefficient of

x^{5}

(1 + x^{2})^{5} (1 + x)^{4}

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