Question

Multiply: x² + y² + z² − xy + xz + yz by x + y − z

Answer 1

The given expression is

 ${\displaystyle x^2+y^2+z^2-xy+xz+yz}$

We have to multiply the above expression by ${\displaystyle (x+y-y)}$.

The required product is

${\displaystyle (x+y-z)(x^2+y^2+z^2-xy+xz+yz)}$

${\displaystyle =\{x+y+(-z)\}\{{\left(x\right)}^2+\left(y^2\right)+{(-z)}^2=-x,y-y(-z)-(-z).x\}}$

Recall the formula

${\displaystyle a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)}$

Using the above formula, we have

${\displaystyle =x^3+y^3+{(-z)}^3-3.x.y.(-z)}$

${\displaystyle =x^3+y^3-z^3+3xyz}$