Question

complex numbers & higher order polynomials practice
complete this assessment to review what you’ve learned. it will not count toward your grade.
one of the solutions to the equation $x^3 - 5x^2 + 6x = 0$ is 2. find the full solution set. which of the following options correctly describes the solution set?
option #1. it has one real solution and two non - real solutions.
option #2. it has one non - real solution and two real solutions.
option #3. it has three real solutions.
(1 point)
option #\square

Explanation:

Step1: Factor the equation

Given the equation \(x^{3}-5x^{2}+6x = 0\), we can factor out an \(x\) first. So we get \(x(x^{2}-5x + 6)=0\).

Step2: Factor the quadratic

Now, we factor the quadratic expression \(x^{2}-5x + 6\). We need two numbers that multiply to \(6\) and add up to \(- 5\). Those numbers are \(-2\) and \(-3\). So, \(x^{2}-5x + 6=(x - 2)(x - 3)\).

Step3: Find the solutions

Substituting back into the factored form of the equation, we have \(x(x - 2)(x - 3)=0\). Using the zero - product property (if \(ab = 0\), then either \(a = 0\) or \(b = 0\)), we set each factor equal to zero:

• If \(x=0\), then the equation is satisfied.
• If \(x - 2=0\), then \(x = 2\).
• If \(x - 3=0\), then \(x = 3\).

All three solutions \(x = 0\), \(x = 2\), and \(x = 3\) are real numbers.

Answer:

Option #3. It has three real solutions.