Question

Sketch a diagram and expand the polynomial geometrically: $(x+3y+2z{)}^2$

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 $=$ $(x+3y+2z{)}^2$
Step 4: Now we have to find the area of $3$ inside square(red, yellow, green) $=$ $x^2+(3y{)}^2+(2z{)}^2$
Step 5: Consider the area of $2$ pink rectangle $=$ length $\times$ breadth $=$ $3xy+3xy=6xy$
Step 6: Area of $2$ purple rectangle $=$ $2xz+2xz=4xz$ and Area of $2$ blue rectangle $=$ $6yz+6yz=12yz$
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., $(x+3y+2z{)}^2=x^2+(5y{)}^2+(2z{)}^2+6xy+4xz+12yz$
Hence, geometrically we expanded the identity $(x+3y+2z{)}^2=x^2+(5y{)}^2+(2z{)}^2+6xy+4xz+12yz$.