(x)(x-5)(2x+3)

Jasonbradley

3 min read Jun 16, 2024

Factoring and Solving the Equation: (x)(x-5)(2x+3) = 0

This expression represents a cubic polynomial in factored form. Let's break down what it means and how to solve it.

Understanding the Expression

(x)(x-5)(2x+3) is a product of three factors. Each factor represents a linear expression.
Linear expression: A linear expression is an expression where the highest power of the variable is 1. In this case, we have:
- x: This is a simple linear expression.
- (x-5): This is another linear expression.
- (2x+3): This is a linear expression with a coefficient of 2 for the 'x' term.

Finding the Roots (Solutions)

To find the values of 'x' that make the entire expression equal to zero, we use the Zero Product Property. This property states:

If the product of two or more factors is zero, then at least one of the factors must be zero.

Applying this to our expression:

x = 0: The first factor, 'x', will be zero when x = 0.
x - 5 = 0: Solving for x, we get x = 5.
2x + 3 = 0: Solving for x, we get x = -3/2.

The Solutions

Therefore, the solutions or roots of the equation (x)(x-5)(2x+3) = 0 are:

x = 0
x = 5
x = -3/2

These are the values of 'x' that make the entire expression equal to zero.

Visualizing the Solution

The equation (x)(x-5)(2x+3) = 0 represents a cubic function. The solutions we found correspond to the points where the graph of this function crosses the x-axis. In other words, these are the x-intercepts of the graph.

Conclusion

By understanding factoring and the Zero Product Property, we can easily solve equations like (x)(x-5)(2x+3) = 0 and find the values of 'x' that make the expression equal to zero. This is a fundamental concept in algebra with applications in many areas of mathematics and beyond.