Let's create an identity

Algebra Level 5

$\begin{aligned} w &=& { a }^{ 4 }+{ b }^{ 4 }+{ c }^{ 4 }+{ d }^{ 4 } \\ x &=& { a }^{ 2 }+b^{ 2 }+{ c }^{ 2 }+{ d }^{ 2 } \\ y &=& ab+ac+ad+bc+bd+cd \\ z &=& abcd \\ \end{aligned}$

Let $a,b,c$ and $d$ be real numbers; and $w,x,y$ and $z$ some unknown constants such that they satisfy the system of equations above. State the following expression

$(a+b+c+d)({ a }^{ 3 }+{ b }^{ 3 }+{ c }^{ 3 }+{ d }^{ 3 }+abc+abd+bcd+acd)$

in terms of $w,x,y$ and $z$ .

4(xy+wz)

{ (w+x) }^{ 3 }-yz

(x+y-1)(w-{ z }^{ 2 })

wx+({ y-z })^{ 3 }

w-x+yz

w+xy+4z

w+y-x{ z }^{ 2 }

{ (w+x+y+z) }^{ 2 }