Question

Simplify each of the following expressions:

(i) $(x+y+z{)}^2+{\left(x+\frac{y}{2}+\frac{z}{3}\right)}^2-{\left(\frac{x}{2}+\frac{y}{3}+\frac{z}{4}\right)}^2$

(ii) (x + y− 2z)² − x² − y² − 3z² + 4xy

(iii) (x²− x + 1)² − (x² + x + 1)²

Answer 1

In the given problem, we have to simplify the value of each expression

(i) Given

We shall use the identity for each bracket

$-\left\{{\left(\frac{x}{2}\right)}^2+{\left(\frac{y}{3}\right)}^2+{\left(\frac{z}{4}\right)}^2+2\left(\frac{x}{2}\right)\left(\frac{y}{3}\right)+2\left(\frac{y}{3}\right)\left(\frac{z}{4}\right)+2\left(\frac{x}{2}\right)\left(\frac{z}{4}\right)\right\}$

By arranging the like terms we get

Now adding or subtracting like terms,

Hence the value of is

(ii) Given

We shall use the identity ${\left(x+y-z\right)}^2=x^2+y^2+z^2+2xy-2yz-2zx$ for expanding the brackets

Now arranging liked terms we get,

Hence the value of is

(iii) Given

We shall use the identity for each brackets

$$\begin{gathered} {\left(x^2-x+1\right)}^2-{\left(x^2+x+1\right)}^2=\left[{\left(x^2\right)}^2+{\left(-x\right)}^2+1^2-2x^3-2x+2x^2\right] \\ -\left[{\left(x^2\right)}^2+{\left(x\right)}^2+1^2+2x^3+2x+2x^2\right] \end{gathered}$$

Canceling the opposite term and simplifies

Hence the value of is .