Fundamental Concepts

84. The Area Model of Division · Visualizing polynomial division with rectangles

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

6x²4x15x102x+53x+2width = 425width = 425height = 375height = 375X
Polynomial division undoes multiplication. To divide a polynomial by a binomial, arrange its area as a rectangle with one side equal to the divisor — the other side is the quotient. Splitting the rectangle into four sub-rectangles and matching the areas reveals each term of the quotient.

Stuck? Ask Guru

For subscribers · Selina ICSE: Fundamental Concepts

What this lesson covers

Try to break it

Set x to 0. Notice how the terms with x disappear, leaving only the constant area.

How you build it

Construct the area model for the division.

  • 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 four sub-areas add up to the area of the whole rectangle.

  • The total width is 3x + 2.
  • The total height is 2x + 5.
  • The area of the top-left rectangle is 6x².
  • The area of the top-right rectangle is 4x.
  • The area of the bottom-left rectangle is 15x.
  • The area of the bottom-right rectangle is 10.
  • Summing the areas gives 6x² + 19x + 10.
  • Thus, (6x² + 19x + 10) ÷ (3x + 2) = 2x + 5.

Worked example

Divide 6x² + 19x + 10 by 3x + 2 using the area model. What is the quotient?

Using the area model, we find the dimensions are 3x+2 and 2x+5. Since one dimension is the divisor, the other is the quotient: 2x+5.

  • 2x + 5 — correct
  • 2x + 10
  • 3x + 5
  • 6x + 2
Chat with us

Subscription Status