Question

How do I factor (x+1)^3/2 - x^2(x+1)^1/2 ?

Answer 1

`(3)/(2)=1(1)/(2)=1+(1)/(2)\\\\(x+1)^(3)/(2)-x^2(x+1)^(1)/(2)=(x+1)^{1+(1)/(2)}-x^2(x+1)^(1)/(2)\\\\=(x+1)^1\cdot(x+1)^(1)/(2)-x^2(x+1)^(1)/(2)=(x+1)(x+1)^(1)/(2)-x^2(x+1)^(1)/(2)\\\\=(x+1)^(1)/(2)(x+1-x^2)\\\\Answer:\boxed{(x+1)^(1)/(2)(-x^2+x+1)}\\\\If\ you\ want\ factor\ completely:\\\\-x^2+x+1=0\\a=-1;\ b=1;\ c=1\\\Delta=b^2-4ac\\\Delta=1^2-4\cdot(-1)\cdot1=1+4=5 \ \textgreater \ 0\\\\x_1=(-b-\sqrt\Delta)/(2a)\ and\ x_2=(-b+\sqrt\Delta)/(2a)`


`x_1=(-1-\sqrt5)/(2\cdot1)=(-1-\sqrt5)/(2)\ and\ x_2=(-1+\sqrt5)/(2)\\\\\boxed{-(x+1)^(1)/(2)\left(x-(-1-\sqrt5)/(2)\right)\left(x-(-1+\sqrt5)/(2)\right)}`