Question

Question: Let x and y be real numbers. Find the coefficient of ${\mathit{x}}^{\mathbf{5}}{\mathit{y}}^{\mathbf{8}}$in the expansion of ${\left(2x-y^2\right)}^{\mathbf{9}}$. [Hint: Apply Theorem E.1 with a = 2x, b = y².]

Short answer

Answer:

The coefficient of${\mathit{x}}^{\mathbf{5}}{\mathit{y}}^{\mathbf{8}}$ in the expansion of ${\left(2x-y^2\right)}^9$is 4032 .

Step-by-step solution

Definition of the Binomial Theorem 

This theorem states that let R be a commutative rind and $\mathit{a}\mathbf{,}\mathit{b}\mathbf{\in }\mathit{R}$. Then each positive integer n,

${\left(a+b\right)}^{\mathbf{n}}\mathbf{=}{\mathit{a}}^{\mathbf{n}}\mathbf{+}\left[\begin{array}{c}n\\ 1\end{array}\right]{\mathit{a}}^{\mathbf{n}\mathbf{-}\mathbf{1}}\mathit{b}\mathbf{+}\left[\begin{array}{c}n\\ 2\end{array}\right]{\mathit{a}}^{\mathbf{n}\mathbf{-}\mathbf{2}}{\mathit{b}}^{\mathbf{2}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}\left[\begin{array}{c}n\\ n-1\end{array}\right]\mathit{a}{\mathit{b}}^{\mathbf{n}\mathbf{-}\mathbf{1}}\mathbf{+}{\mathit{b}}^{\mathbf{n}}$

Find the coefficient of x5y8 in the expansion of (2x-y2)9 

Consider the equation as follows:

${\left(2x-y^2\right)}^9$

Let a = 2x and $b=y^2$.

Apply the binomial theorem and substitute the corresponding values.

Reduce the above expression as follows.

$$\begin{gathered} {\left(2x-y^2\right)}^9=\left[512x^9+2304x^8{y}^2+4608x^7{y}^4+5376x^6{y}^6+4032x^5{y}^8+...+ \\ 2016x^4{y}^{10}+672x^3{y}^{12}+144x^2{y}^{14}+18x{y}^{16}+y^{18} \\ \right] \end{gathered}$$

Therefore, the coefficient of $x^5{y}^8$in the expansion of ${\left(2x-y^2\right)}^9$is4032.