# How do you find the zeroes of f (x) =5x^4 − 2x^2 − 3?

May 3, 2015

This function is an example of a bi-quadratic function, which is the polynomial function of the ${4}^{t h}$ degree with no terms of an odd degree.

The general polynomial of the ${4}^{t h}$ degree looks like this:
$f \left(x\right) = {a}_{0} {x}^{4} + {a}_{1} {x}^{3} + {a}_{2} {x}^{2} + {a}_{3} {x}^{1} + {a}_{4} {x}^{0}$
Since no odd degree terms are present, general expression for a bi-quadratic function is:
$f \left(x\right) = {a}_{0} {x}^{4} + {a}_{2} {x}^{2} + {a}_{4} {x}^{0}$

Finding the values of an unknown $x$ where this function equals to zero is a simple three-step procedure.

Step 1. Substitute $y = {x}^{2}$. Then the equation $f \left(x\right) = 0$ that determines the zeros of a function is transformed into an equation with an unknown $y$:
${a}_{0} {y}^{2} + {a}_{2} y + {a}_{4} = 0$

Step 2. The above equation is a regular quadratic equation that we know how to solve. Its two solutions are:
${y}_{1} = \frac{- {a}_{2} + \sqrt{{a}_{2}^{2} - 4 {a}_{0} {a}_{4}}}{2 {a}_{0}}$
${y}_{2} = \frac{- {a}_{2} - \sqrt{{a}_{2}^{2} - 4 {a}_{0} {a}_{4}}}{2 {a}_{0}}$
(solutions might not be real if ${a}_{2}^{2} - 4 {a}_{0} {a}_{4} < 0$, they are supposed to be discarded).

Step 3. Knowing the value of an unknown $y$ (actually, from zero up to two values, depending on coefficients), we can find up to four values of $x$ since $y = {x}^{2}$:
x_1=sqrt(y_1); x_2=-sqrt(y_1); x_3=sqrt(y_2); x_4=-sqrt(y_2);
(depending on the coefficients, certain solutions might not be real)

I think it would be useful for a student who ask this question to do the math with concrete coefficients given in the problem.

As an illustration, here is a graph of the given function that shows where it takes zero values. It shows that this function has only two real values of $x$ where it equals to zero, $x = 1$ and $x = - 1$, which implies that one of the solutions of an equation for $y$ is negative and there is no $x$ that would be equal to it if raised to a power of $2$.
graph{5x^4-2x^2-3 [-3, 3, -4, 4]}