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How to Factor Using Area Model?

Published in Algebra 3 mins read

Factoring using the area model (also known as the box method) is a visual and organized way to factor quadratic expressions and other polynomials. Here's how to do it:

Steps for Factoring with the Area Model

  1. Set up the Area Model (Box): Draw a 2x2 grid (a box with four sections).

  2. Place the First and Last Terms: Write the first term (usually the $x^2$ term) of the quadratic expression in the upper-left box. Write the last term (the constant term) in the lower-right box.

  3. Find the Factor Pair: Multiply the first and last terms together. Then, find two factors of this product that add up to the middle term (the $x$ term) of the original quadratic expression.

  4. Place the Factor Pair in the Remaining Boxes: Write the two factors you found in the previous step into the remaining two boxes (the upper-right and lower-left boxes). The order in which you place the factors typically doesn't matter.

  5. Find the Greatest Common Factor (GCF) of Each Row and Column: Determine the greatest common factor for each row and each column of the area model. Write these GCFs outside the box, along the top and left sides. Remember to include the sign (+ or -) when writing the GCFs.

  6. Write the Factored Expression: The GCFs you wrote outside the box represent the factors of the original quadratic expression. Write them as two binomials in parentheses.

Example

Let's factor the expression $x^2 + 5x + 6$ using the area model:

  1. Set up the box:

  2. Place First and Last Terms:

    $x^2$
    6

  3. Find the Factor Pair: $x^2 * 6 = 6x^2$. We need two factors of $6x^2$ that add up to $5x$. Those factors are $2x$ and $3x$.

  4. Place the Factor Pair:

    $x^2$ $3x$
    $2x$ 6

  5. Find the GCFs:

    • Row 1: GCF of $x^2$ and $3x$ is $x$
    • Row 2: GCF of $2x$ and $6$ is $2$
    • Column 1: GCF of $x^2$ and $2x$ is $x$
    • Column 2: GCF of $3x$ and $6$ is $3$

    Area Model with GCFs:

    $x$ $3$
    $x$ $x^2$ $3x$
    $2$ $2x$ 6

  6. Write the Factored Expression: The factors are $(x + 2)$ and $(x + 3)$. Therefore, $x^2 + 5x + 6 = (x + 2)(x + 3)$.

Key Considerations

  • Signs: Pay close attention to the signs of the terms, especially when finding the factor pair and the GCFs.
  • Practice: The area model becomes easier with practice. Work through several examples to get comfortable with the process.
  • Leading Coefficient: If the quadratic has a leading coefficient other than 1 (e.g., $2x^2 + 5x + 2$), the process is the same, but the numbers involved are often larger, and finding the correct factor pair can be more challenging.
  • Not Factorable: Some quadratic expressions cannot be factored using integers. In these cases, the area model will not lead to a simple factored form.

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