Question

What is the expansion of the polynomial: $(3x+2y+z{)}^2$?

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