Question

Expand the polynomial identity geometrically:$(p+q^2+2r{)}^2$

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