Fundamental Concepts

83. Area of the Whole · Visualizing polynomial multiplication

The total area equals the sum of the four sub-rectangle areas.

w₁w₂h₁h₂(w₁+w₂)(h₁+h₂)w₁h₁+w₂h₁+w₁h₂+w₂h₂w₁ = 200w₁ = 200w₂ = 300w₂ = 300h₁ = 150h₁ = 150h₂ = 200h₂ = 200w₁h₁ = 30000w₁h₁ = 30000w₂h₁ = 45000w₂h₁ = 45000w₁h₂ = 40000w₁h₂ = 40000w₂h₂ = 60000w₂h₂ = 60000Total area = 175000Total area = 175000
Multiplying two binomials (a + b)(c + d) gives four products — ac, ad, bc, bd. The area-model rectangle shows them as four sub-areas whose sum equals the total area (a+b)(c+d).

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For subscribers · Selina ICSE: Fundamental Concepts

What this lesson covers

Try to break it

Drag the handles on the edges. Watch how the four inner rectangles change size. Notice that the total area is always the sum of the four parts.

How you build it

Build the area model of (a+b)(c+d).

  • Place point A as the bottom-left corner.
  • Place point C above and to the right of A, as the top-right corner.
  • Draw the rectangle from corner A to corner C.
  • Place point P on the bottom side, between the two bottom corners.
  • Draw a vertical segment from P straight up to the top side.
  • Place point R on the left side, between the two left corners.
  • Draw a horizontal segment from R straight across to the right side.

The proof, step by step

Prove that the total area equals the sum of the four sub-rectangle areas.

  • Total width is w₁ + w₂. Total height is h₁ + h₂.
  • Total Area = (w₁ + w₂)(h₁ + h₂).
  • Area of R1 = w₁h₁, R2 = w₂h₁, R3 = w₁h₂, R4 = w₂h₂.
  • Sum of areas = w₁h₁ + w₂h₁ + w₁h₂ + w₂h₂.
  • Factor sum: w₁(h₁+h₂) + w₂(h₁+h₂) = (w₁+w₂)(h₁+h₂).

Worked example

A rectangle has length (2x + 3) and width (x + 4). What is the area in terms of x?

Expand (2x+3)(x+4) = 2x² + 8x + 3x + 12 = 2x² + 11x + 12.

  • 2x² + 11x + 12 — correct
  • 2x² + 5x + 12
  • 2x² + 11x + 7
  • 2x² + 8x + 12
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