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.
Answer and Explanation: 1
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}
Learn more about this topic:
from
Chapter 20 / Lesson 3Learn about polynomial equations and polynomial formulas. Understand how to find the roots of a polynomial and how to generate polynomials from roots.