Question

Use the binomial theorem to expand the expression:(x+3y)^4

Answer 1

We will solve as follows:

$(x+3y)^4=(x+3y)^2\cdot(x+3y)^2$$=(x^2+6xy+9y^2)\cdot(x^2+6xy+9y^2)$$=x^2(x^2+6xy+9y^2)+6xy(x^2+6xy+9y^2)+9y^2(x^2+6xy+9y^2)$$=(x^4+6x^3y+9x^2y^2)+(6x^3y+36x^2y^2+54xy^3)+(9x^2y^2+54xy^3+81y^4)$$=x^4+(6x^3y+6x^3y)+(9x^2y^2+36x^2y^2+9x^2y^2)+(54xy^3+54xy^3)+81y^4$$=x^4+12x^3y+54x^2y^2+108xy^3+81y^4$

So, the expansion for the binomial is:

$(x+3y)^4=x^4+12x^3y+54x^2y^2+108xy^3+81y^4$