Question

Verify the polynomial geometrically: $(x+y+z{)}^2$

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 $=$ $(x+y+z{)}^2$
Step 4: Now we have to find the area of $3$ inside square(red, yellow, green) $=$ $x^2+y^2+z^2$
Step 5: Consider the area of $2$ pink rectangle $=$ length $x$ breadth $=$ $xy+xy=2xy$
Step 6: Area of $2$ purple rectangle $=$ $xz+xz=2xz$ and Area of $2$ blue rectangle $=$ $yz+yz=2yz$
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., $(x+y+z{)}^2=x^2+y^2+z^2+2xy+2yz+2xz$
Hence, geometrically we proved the identity $(x+y+z{)}^2=x^2+y^2+z^2+2xy+2yz+2xz$.