Question

Explain the polynomial geometrically: $(x-2z)(x+2z)=x^2-(2z{)}^2$.

Answer 1

Step 1: Draw a square and cut into $3$ parts.
Step 2: There are $1$ hided square green and $2$ rectangles (pink, blue)
Step 3: Area of the full square $=$ $(x{)}^2-(2z{)}^2$
Step 4: Now we have to find the area of rectangle as shown in the figure.
Step 5: Consider the area of pink rectangle $=$ length $\times$ breadth $=$ $x(x-2z)$
Step 6: Area of blue rectangle $=$ $2z(x-2z)$
Step 7: Area of full square $=$ area of pink rectangle $+$ area of blue rectangle.
i.e., $(x{)}^2-(2z{)}^2=x(x-2z)+2z(x-2z)$
$(x{)}^2-(2z{)}^2=(x+2z)(x-2zy)$
Hence, geometrically we proved the identity $(x{)}^2-(2z{)}^2=(x+2z)(x-2zy)$.