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What is the coefficient of (a) x^10. (b) x^11. In the expansion of (2 x - 1 / x^2)^{16}? Fully...

Question:

What is the coefficient of

(a) {eq}x^{10} {/eq}.

(b) {eq}x^{11} {/eq}.

In the expansion of {eq}\bigg(2 x - \dfrac 1 {x^2}\bigg)^{16} {/eq}? Fully evaluate your answers.

Binomial Theorem:


The binomial theorem is used to expand the higher order polynomials. It is also used in the mass function of the binomial distribution. That is why, the name of the distribution is binomial. The combination method is involved in the theorem.

Answer and Explanation:


Given Information:

The binomial theorem is:

{eq}{\left( {a + b} \right)^n} = \sum\limits_{r = 0}^n {{}^n{C_r}{{\left( a \right)}^r}{{\left( b \right)}^{n - r}}} {/eq}

The following expression can be expanded using Binomial theorem:

{eq}\begin{align*} {\left( {2x - \dfrac{1}{{{x^2}}}} \right)^{16}} &= \sum\limits_{r = 0}^{16} {{}^{16}{C_r}{{\left( {2x} \right)}^r}{{\left( { - {x^{ - 2}}} \right)}^{16 - r}}} \\ &= {}^{16}{C_0}{\left( {2x} \right)^0}{\left( { - {x^{ - 2}}} \right)^{16}} + {}^{16}{C_1}\left( {2x} \right){\left( { - {x^{ - 2}}} \right)^{15}} + ............ + {}^{16}{C_{16}}{\left( {2x} \right)^{16}}{\left( { - {x^{ - 2}}} \right)^{16 - 16}}\\ &= 65536{x^{16}} - 524288{x^{13}} + 1966080{x^{10}} - 4587520{x^7} + 7454720{x^4} - 8945664x + \dfrac{{8200192}}{{{x^2}}} - \dfrac{{5857280}}{{{x^5}}} + \dfrac{{3294720}}{{{x^8}}} - \dfrac{{1464320}}{{{x^{11}}}} + \dfrac{{512512}}{{{x^{14}}}} - \dfrac{{139776}}{{{x^{17}}}} + \dfrac{{29120}}{{{x^{20}}}} - \dfrac{{4480}}{{{x^{23}}}} + \dfrac{{480}}{{{x^{26}}}} - \dfrac{{32}}{{{x^{29}}}} + \dfrac{1}{{{x^{32}}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 1 \right) \end{align*} {/eq}

a)

The coefficient of {eq}{x^{10}}\,\,in\,\,the\,\,\exp ansion\,\,of\,\,{\left( {2x - \dfrac{1}{{{x^2}}}} \right)^{16}} {/eq} is 1966080. This is obtained from (1).

b)

The coefficient of {eq}{x^{11}}\,\,in\,\,the\,\,\exp ansion\,\,of\,\,{\left( {2x - \dfrac{1}{{{x^2}}}} \right)^{16}} {/eq} is 0. This is obtained from (1).


Learn more about this topic:

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The Binomial Theorem: Defining Expressions

from Algebra II: High School

Chapter 12 / Lesson 7
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