Find the real roots of {eq}x^4 = 16 {/eq}.

Question:

Find the real roots of {eq}x^4 = 16 {/eq}.

Roots of polynomials:

The roots of a polynomial {eq}P(x) {/eq} are the solutions to the equation

$$P(x)=0$$

Techniques for solving this equation include factoring, using the quadratic equation, and the rational roots theorem.

Here, we have to find the real roots of {eq}x^{4}-16. {/eq} That is, we have to solve the equation

$$x^4-16=0.$$

Factoring, the polynomial, we get:

\begin{align} x^4-16&=0 \\ (x^2-4)(x^2+4)&=0 \\ (x-2)(x+2)(x^2+4)&=0 \end{align}

Therefore, the equation is true if {eq}x-2=0 {/eq} (that is, if {eq}x=2) {/eq} or if {eq}x+2=0 {/eq} (that is, if {eq}x=-2) {/eq} or if {eq}x^2+4=0 {/eq} (that is, if {eq}x^2=-4). {/eq} This last equation has no real roots, so the only real roots of {eq}x^{4}=16 \text{ are }\boxed{+2, -2}. {/eq}