(x-4)(x+3)(x-5)

3 min read Jun 17, 2024

Factoring and Solving the Expression (x - 4)(x + 3)(x - 5)

The expression (x - 4)(x + 3)(x - 5) represents a polynomial in factored form. This form provides valuable insights into the polynomial's roots, behavior, and graph.

Understanding the Factored Form

Roots: The factored form directly reveals the roots (or zeros) of the polynomial. Setting each factor equal to zero and solving gives us the roots:
- x - 4 = 0 => x = 4
- x + 3 = 0 => x = -3
- x - 5 = 0 => x = 5
Multiplicity: Each root appears only once, meaning the multiplicity of each root is 1. This implies that the graph will cross the x-axis at each root.
Degree: The expression is a product of three linear factors, indicating a polynomial of degree 3. This means it has a maximum of three roots and its graph will have a general "S" shape.

Expanding the Expression

To obtain the polynomial in standard form, we can expand the product:

Start with the first two factors: (x - 4)(x + 3) = x² - x - 12
Multiply the result by the third factor: (x² - x - 12)(x - 5) = x³ - 6x² - 7x + 60

Therefore, the expanded form of the expression is x³ - 6x² - 7x + 60.

Visualizing the Polynomial

The graph of the polynomial will have the following key features:

Intercepts: It will intersect the x-axis at x = 4, x = -3, and x = 5.
End Behavior: Since the leading coefficient is positive and the degree is odd, the graph will rise to the right and fall to the left.

The graph will be a smooth curve that passes through these points and exhibits the characteristic "S" shape of a cubic polynomial.

Conclusion

The factored form (x - 4)(x + 3)(x - 5) provides a concise representation of a cubic polynomial with roots at x = 4, x = -3, and x = 5. Understanding the factored form allows us to easily identify the roots, multiplicity, and degree of the polynomial, and provides insights into the behavior and shape of its graph.