Chapter 9.5, Problem 32E

To determine

The expand and simplify the expression ${\left(2x-5y\right)}^5$.

Answer to Problem 32E

  $32x^5-400x^{4}y+2000x^3{y}^2-5000x^2{y}^3+6250x{y}^4-3125y^5$

Explanation of Solution

Given: ${\left(2x-5y\right)}^5$

FORMULA USED: ${(x+y)}^n=x^n+n{x}^{n-1}y+{\mathrm{.....}}_n{C}_r{x}^{n-r}{y}^r+\mathrm{....}nx{y}^{n-1}+y^n$

The expansion as follows:

  $\begin{array}{c}{\left(2x-5y\right)}^5=C{}_0{\left(2x\right)}^5{\left(-5y\right)}^0+C{}_1{\left(2x\right)}^4{\left(-5y\right)}^1+C{}_2{\left(2x\right)}^3{\left(-5y\right)}^2+C{}_3{(}^2{\left(-5y\right)}^3+C{}_4{\left(2x\right)}^1{\left(-5y\right)}^4+C{}_0{\left(2x\right)}^0{\left(-5y\right)}^5\\ =1\cdot 32x^5+5\cdot 16x^4\cdot \left(-5y\right)+10\cdot 8x^3\cdot 25y^2+10\cdot 4x^2\cdot \left(-125y^3\right)+5\cdot 2x\cdot 625y^4+1\cdot 1\cdot \left(-3125y^5\right)\\ =32x^5-400x^{4}y+2000x^3{y}^2-5000x^2{y}^3+6250x{y}^4-3125y^5\end{array}$

Hence, the expansion is $32x^5-400x^{4}y+2000x^3{y}^2-5000x^2{y}^3+6250x{y}^4-3125y^5$