Look for a quadratic equation that has no real solutions and multiply both sides of it by x.
Example Equation: x^3+x = 0
Practice makes perfect
We have to find a counterexample to the following statement.
A polynomial equation of degree three
always has three real solutions.
We need to find a polynomial of degree three that has less than three solutions. We can start by considering a quadratic equation that has no real solutions. Below we write one example equation.
x^2+1 = 0
There is no real solution to the equation above. However, it is a quadratic equation. We can obtain a third degree equation by multiplying it by x.
x(x^2+1) = x(0) ⇒
x^3+x = 0
Notice that the final equation can be factored as x(x^2+1)=0. Then, by the Zero Product Property we can set the following two equations.
x(x^2+1) = 0
↙ ↘
x=0 x^2+1 = 0
Since the right-hand equation has no solution, we can conclude that the unique solution to the third degree equation is x=0. Therefore, this is the counterexample we were asked for.
Polynomial Equation
Nº of Solutions
x^3+x = 0
1
Keep in mind that this is just an example equation and so your answer may vary.
Extra
Extra
To see why x^2+1=0 has no real solutions, let's solve this equation for x.
