Understanding the Equation
The equation in question is a cubic polynomial represented as ( x^3 + 2x = 0 ). This expression requires finding the values of ( x ) that satisfy it, commonly referred to as the "roots" of the equation. To solve this, we can factor the cubic and find the values of ( x ) that make the equation true.
Factoring the Equation
To simplify the equation, we can start by factoring out the common term. The equation can be rewritten as follows:
[x(x^2 + 2) = 0
]
This factored form indicates that the product of ( x ) and ( (x^2 + 2) ) equals zero. According to the zero-product property, at least one of the factors must equal zero for the entire product to be zero.
Solving for Roots
-
First factor: ( x = 0 )
The first factor gives us a straightforward solution:
[
x = 0
] -
Second factor: ( x^2 + 2 = 0 )
To explore the second factor, we need to solve for ( x ):
[
x^2 + 2 = 0
] Rearranging gives us:
[
x^2 = -2
]Taking the square root of both sides leads to:
[
x = \pm i\sqrt{2}
] Here, ( i ) represents the imaginary unit, which is defined as ( \sqrt{-1} ). This indicates that the solutions from this factor are complex numbers.
Summary of Roots
Collectively, the roots of the equation ( x^3 + 2x = 0 ) are:
- ( x = 0 ) (real root)
- ( x = i\sqrt{2} ) (complex root)
- ( x = -i\sqrt{2} ) (complex root)
Graphical Interpretation
Visualizing the function ( f(x) = x^3 + 2x ) can provide insight into the nature of the roots. The graph will intersect the x-axis where the real roots are located. In this case, ( x = 0 ) marks a point of intersection. The nature of the other roots indicates that there will not be additional intersections since the cubic polynomial can only have one real root and the rest must be complex.
Analyzing the Solutions
Determining the state of the roots helps in understanding the behavior of the function. The presence of a single real root and two complex roots points to unique characteristics of the graph. The real root contributes to the overall shape, while the complex roots indicate that the function does not touch the x-axis anymore beyond this point. This emphasizes that the polynomial can exhibit local extrema without necessarily crossing the axis multiple times.
FAQ
What are the steps to find the roots of a cubic equation?
To find the roots of a cubic equation, rewrite it in standard form followed by factoring if possible. Set each factor to zero and solve for ( x ). If required, use synthetic division or the quadratic formula for more complex factors.
What does it mean if a root is complex?
A complex root suggests that the polynomial does not intersect the x-axis at those values. Complex roots often come in conjugate pairs and indicate behaviors of the polynomial that cannot be represented with real numbers alone.
Can all cubic equations be factored easily?
Not all cubic equations can be factored easily. Some may require numerical methods or specialized formulas like Cardano’s method for finding roots, especially when no rational roots are apparent.