(n-2)(n^2+n-5)

3 min read Jun 16, 2024

Factoring the Expression: (n-2)(n^2+n-5)

The expression (n-2)(n^2+n-5) is already in factored form. However, we can analyze it further and understand its implications.

Understanding the Factored Form

(n-2): This represents a linear factor. It indicates that the expression will equal zero when n = 2.
(n^2+n-5): This represents a quadratic factor. It doesn't factor further using real numbers. This means it has no real roots, and its graph doesn't intersect the x-axis.

Finding the Roots

To find the roots of the entire expression, we set it equal to zero and solve:

(n-2)(n^2+n-5) = 0

This equation is satisfied when either:

n - 2 = 0 => n = 2
n^2 + n - 5 = 0

The quadratic equation doesn't have easy integer solutions. We can solve it using the quadratic formula:

n = (-b ± √(b^2 - 4ac)) / 2a

Where a = 1, b = 1, and c = -5. Plugging these values into the formula will give us two complex roots.

Applications

This factored expression can be used in various contexts:

Solving equations: Setting the expression equal to zero allows us to find the values of n that make the expression equal to zero.
Graphing functions: The factored form can help us identify the x-intercepts (where the graph crosses the x-axis) and the behavior of the graph near those points.
Finding the domain of a function: The expression can represent the denominator of a rational function. We need to exclude the values of n that make the denominator zero to find the function's domain.

Conclusion

The expression (n-2)(n^2+n-5) is already in factored form. It represents a cubic function with one real root (n = 2) and two complex roots. Understanding the factored form allows us to analyze the function's behavior and use it in various mathematical applications.