(x+1)(x-2)(x+3)

4 min read Jun 16, 2024

Exploring the Expression (x+1)(x-2)(x+3)

This expression represents the product of three linear factors: (x+1), (x-2), and (x+3). Let's delve into what this expression tells us about its corresponding polynomial function.

Expanding the Expression

To understand the polynomial function represented by this expression, we can expand it using the distributive property (often referred to as FOIL for binomial multiplications).

Multiply the first two factors: (x+1)(x-2) = x² - 2x + x - 2 = x² - x - 2
Multiply the result by the third factor: (x² - x - 2)(x+3) = x³ + 3x² - x² - 3x - 2x - 6 = x³ + 2x² - 5x - 6

Therefore, the expanded form of the expression is x³ + 2x² - 5x - 6.

Identifying Key Features

This expanded form reveals important features of the polynomial function:

Degree: The highest power of x is 3, so the degree of the polynomial is 3. This indicates it's a cubic function.
Leading coefficient: The coefficient of the x³ term is 1, which means the function has a positive leading coefficient.
Constant term: The constant term is -6, indicating the y-intercept of the function's graph is at (0, -6).

Roots and Zeros

The factors in the original expression reveal the roots or zeros of the polynomial function. These are the values of x where the function equals zero.

(x+1) = 0 => x = -1
(x-2) = 0 => x = 2
(x+3) = 0 => x = -3

The polynomial function has three roots: -1, 2, and -3.

Graphing the Function

Knowing the roots and the degree of the polynomial helps us sketch its graph. Cubic functions generally have an "S" shape.

The function crosses the x-axis at x = -3, x = -1, and x = 2.
As the leading coefficient is positive, the graph rises to the right (as x approaches positive infinity).
Since it's a cubic function, it will have a turning point between each pair of roots.

Applications

Understanding this expression and its corresponding polynomial function is crucial in various fields:

Calculus: Finding derivatives and integrals of polynomial functions is fundamental in calculus.
Physics: Polynomial functions model various physical phenomena like projectile motion and oscillations.
Engineering: Polynomial functions are used in designing structures, circuits, and algorithms.

Conclusion

The expression (x+1)(x-2)(x+3) represents a cubic polynomial function with roots at -3, -1, and 2. By understanding its expanded form, roots, and general behavior, we gain valuable insights into its characteristics and applications.