Question

Determine the expansion of the polynomial geometrically: $(2m+n^2+l{)}^2$

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