Question

Find the expansion of the polynomial $(2x^2+x+1{)}^2$ geometrically.

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 $=$ $(2x^2+x+1{)}^2$
Step 4: Now we have to find the area of $3$ inside square(red, yellow, green) = $(2x^2{)}^2+x^2+1^2$
Step 5: Consider the area of $2$ pink rectangle $=$ length $\times$ breadth $=$ $2x^3+2x^3=4x^3$

Step 6: Area of $2$ purple rectangle $=$ $2x^2+2x^2=4x^2$ and Area of $2$ blue rectangle $=$ $x+x=2x$
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., $(2x^2+x+1{)}^2=(2x^2{)}^2+x^2+1^2+4x^3+4x^2+2x$
Hence, geometrically we proved the identity $(2x^2+x+1{)}^2=(2x^2{)}^2+x^2+1^2+4x^3+4x^2+2x$