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(2x-3)(x+6)(x-4)

4 min read Jun 16, 2024
(2x-3)(x+6)(x-4)

Factoring and Expanding the Expression (2x-3)(x+6)(x-4)

This expression represents the product of three linear factors. To fully understand its behavior, we can explore both its factored form and its expanded form.

Understanding the Factored Form

The expression is already in factored form, which is extremely useful for several reasons:

  • Finding the roots (or zeros): The roots of a polynomial are the values of 'x' that make the entire expression equal to zero. Since we have the factored form, we can easily see that the roots are:

    • x = 3/2 (from the factor 2x-3)
    • x = -6 (from the factor x+6)
    • x = 4 (from the factor x-4)

  • Identifying the x-intercepts: The roots of a polynomial also correspond to the x-intercepts of its graph. Therefore, the graph of this expression will cross the x-axis at the points (3/2, 0), (-6, 0), and (4, 0).

  • Understanding the behavior of the function: The factored form helps us visualize how the expression changes as 'x' increases. Each factor contributes to the overall value, and we can analyze the sign of each factor for different ranges of 'x'.

Expanding the Expression

While the factored form is convenient, expanding the expression can also provide valuable insights. To expand, we can use the distributive property (often referred to as FOIL) multiple times.

Here's how the expansion goes:

  1. First, multiply the first two factors: (2x-3)(x+6) = 2x² + 9x - 18

  2. Now, multiply the result by the third factor: (2x² + 9x - 18)(x-4) = 2x³ - x² - 54x + 72

Therefore, the expanded form of the expression is 2x³ - x² - 54x + 72.

Applications

Understanding both the factored and expanded forms of this expression has various applications, including:

  • Solving equations: Setting the expanded form equal to zero allows us to find the solutions of the equation.
  • Graphing the function: The expanded form helps us analyze the end behavior and turning points of the graph.
  • Finding the area of a region: The expression could represent the area of a three-dimensional object if each factor represents a dimension.

In conclusion, understanding both the factored and expanded forms of (2x-3)(x+6)(x-4) provides valuable insights into its behavior and allows us to solve various problems related to polynomials.

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