Question

Find the expansion of the polynomial identity used in the diagram geometrically.

• A. $(4a-b+c{)}^2=(4a{)}^2+b^2+c^2+8ab+8ac+2bc$
• B. $(4a+b+c{)}^2=(4a{)}^2+b^2+c^2-8ab-4ac-2bc$
• C. $(4a+b+c{)}^2=(4a{)}^2+b^2+c^2+8ab+8ac+2bc$
• D. $(4a+b+c{)}^2=(4a{)}^2+b^2+c^2+4ab+4ac+2bc$

Answer 1

The correct option is C $(4a+b+c{)}^2=(4a{)}^2+b^2+c^2+8ab+8ac+2bc$

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 = $(4a+b+c{)}^2$
Step 4: Now we have to find the area of $3$ inside square(red, yellow, green) $=$ $(4a{)}^2+b^2+c^2$
Step 5: Consider the area of $2$ pink rectangle $=$ length $\times$ breadth $=$ $4ab+4ab=8ab$
Step 6: Area of $2$ purple rectangle $=$ $4ac+4ac=8ac$ and Area of $2$ blue rectangle $=$ $bc+bc=2bc$
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., $(4a+b+c{)}^2=(4a{)}^2+b^2+c^2+8ab+8ac+2bc$
So, $(4a+b+c{)}^2=(4a{)}^2+b^2+c^2+8ab+8ac+2bc$ is the expansion used in the diagram.