what is (a+b)^3

(a+b)^3 ?

Let a = x:

If  $$(x+b)^3=0$$    then   $$(x+b)(x+b)(x+b) = 0$$   is a Polynom with the roots $$x_1 = -b\qquad x_2 = -b\qquad x_3 = -b$$ .

Vieta's formulas:

If $$x^3+Ax^2+Bx+C=0$$ , then

$$\\\small{\text{$ A =-(x_1 +x_2 +x_3 )\qquad B =x_1 x_2 +x_1 x_3 +x_2 x_3 \qquad C =-(x_1 x_2 x_3 ) $}}\\\\ \small{\text{$ \begin{array}{rcl} A &=& -[ (-b) +(-b) +(-b) ] = -[3(-b)]=3b\\ B &=& (-b) (-b) +(-b) (-b) +(-b) (-b)=3b^2 \\ C &=& -[(-b) (-b) (-b) ]=-[-3b^3]=b^3 \end{array} $}}\\$$

so we have:

$$\\ (x+b)^3= x^3+Ax^2+Bx+C=0 \\ (x+b)^3= x^3+(3b)x^2+(3b^2)x+(b^3) \qquad | \qquad \textcolor[rgb]{1,0,0}{x = a}\\ (a+b)^3= a^3+(3b)a^2+(3b^2)a+(b^3)\\\\ \mathbf{(a+b)^3= a^3+3a^2b+3ab^2+b^3}$$

It can be expanded as:

$$(a + b)^3=a^3+3a^2b+3ab^2+b^3$$

.