Question

How do you expand `(3x-2y)^5`?

Answer 1

`243x^5-810x^4y+1080x^3y^2-720x^2y^3+240xy^4-32y^5`

Explanation:

`"using the "color(blue)"binomial theorem"`

`•color(white)(x)(x+y)^n=sum_(r=0)^n((n),(r))x^(n-r)y^r`

`"where" ((n),(r))=(n!)/(r!(n-r)!)`

`"here " x=3x" and " y=-2y`

`rArr(3x-2y)^5`

`=((5),(0))(3x)^5(-2y)^0+((5),(1))(3x)^4(-2y)^1+((5),(2))(3x)^3(-2y)^2+((5),(3))(3x)^2(-2y)^3+((5),(4))(3x)^1(-2y)^4+((5),(5))(3x)^0(-2y)^5`

`"we can obtain the binomial coefficients using the appropriate"`
`"row of "color(blue)"Pascal's triangle"`

`"for n = 5 the row of coefficients is"`

`1color(white)(x)5color(white)(x)10color(white)(x)10color(white)(x)5color(white)(x)1`

`=1.243x^5+5.81x^4(-2y)+10.27x^3. 4y^2+10.9x^2(-8y^3)+5.3x.16y^4+1.-32y^5`

`=243x^5-810x^4y+1080x^3y^2-720x^2y^3+240xy^4-32y^5`