Question

Explain and expand the polynomial geometrically: $(2x+2y+2z{)}^2$

Answer 1

Step 1: Draw a square and cut into $9$ parts.
Step 2: There are $3$ squares (red, yellow, green) and $6$ rectangles ($2$ pink, $2$ purple, $2$ blue)
Step 3: Area of the full square $=$ $(2x+2y+2z{)}^2$
Step 4: Now we have to find the area of $3$ inside square(red, yellow, green) $=$ $(2x{)}^2+(2y{)}^2+(2z{)}^2$
Step 5: Consider the area of $2$ pink rectangle $=$ length $\times$ breadth $=$ $4xy+4xy=8xy$
Step 6: Area of $2$ purple rectangle $=$ $4xz+4xz=8xz$ and Area of $2$ blue rectangle $=$ $4yz+4yz=8yz$
Step 7: Area of full square $=$ area of $3$ inside square $+$ area of $2$ pink rectangle $+$ area of $2$ purple rectangle $+$ area of $2$ blue rectangle.
i.e., $(2x+2y+2z{)}^2=(2x{)}^2+(2y{)}^2+(2z{)}^2+8xy+8xz+8yz$
Hence, geometrically we expanded the identity $(2x+2y+2z{)}^2=(2x{)}^2+(2y{)}^2+(2z{)}^2+8xy+8xz+8yz$.