Verified Answer :

  • Identify the binomials \((2x + 3)\) and \((x - 5)\) from the table headers.
  • Sum the grid entries: \(2x^2 - 10x + 3x - 15\).
  • Simplify by combining like terms \(-10x\) and \(3x\) to get \(-7x\).
  • State the final product: \(\boxed{2x^2 - 7x - 15}\).

Explanation

  1. Analyzing the Area Model Grid

To determine the product represented by the area model, we look at the headers along the top and left sides of the grid. The horizontal headers are \(x\) and \(-5\), which combine to form the binomial \((x - 5)\). The vertical headers are \(2x\) and \(+3\), which combine to form the binomial \((2x + 3)\). The area model represents the multiplication of these two binomials: \((2x + 3)(x - 5)\).

  1. Calculating Partial Products

The interior of the grid contains the individual terms that result from multiplying the corresponding headers:

  • The top-left cell: \(2x \times x = 2x^2\)
  • The top-right cell: \(2x \times (-5) = -10x\)
  • The bottom-left cell: \(3 \times x = 3x\)
  • The bottom-right cell: \(3 \times (-5) = -15\)

The product is equal to the sum of all these interior terms: \(2x^2 + (-10x) + 3x - 15\).

  1. Combining Like Terms and Final Result

To simplify the expression, we combine the like terms, which are the terms containing \(x\): \(-10x\) and \(3x\). Adding them together: \(-10x + 3x = -7x\). Therefore, the final expanded form of the product is \(2x^2 - 7x - 15\).

Examples

Area models like this are incredibly useful for visual learners to understand polynomial multiplication. Imagine you are designing a garden plot that is \(2x + 3\) feet long and \(x - 5\) feet wide. To find the total area of this plot, you multiply the length by the width. The area model breaks this down into smaller, manageable rectangles (like a \(2x \times x\) section and a \(3 \times -5\) section), which helps ensure no part of the calculation is missed. This method is the foundation for understanding how to multiply more complex algebraic expressions.

Related Questions