Question

Draw and expand the polynomial geometrically: $(a+b+2c{)}^2$

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 $=$ $(a+b+2c{)}^2$
Step 4: Now we have to find the area of $3$ inside square(red, yellow, green) $=$ $a^2+b^2+(2c{)}^2$
Step 5: Consider the area of $2$ pink rectangle $=$ length $\times$ breadth $=$ $ab+ab=2ab$
Step 6: Area of $2$ purple rectangle $=$ $2ac+2ac=4ac$ and Area of $2$ blue rectangle $=$ $2bc+2bc=4bc$
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., $(a+b+2c{)}^2=a^2+b^2+(2c{)}^2+2ab+4ac+4bc$
Hence, geometrically we proved the identity $(a+b+2c{)}^2=a^2+b^2+(2c{)}^2+2ab+4ac+4bc$.