How do you find all zeros of #f(x)=5x^4+15x^2+10#?

Question

Elijah Allen · Accepted Answer

#f(x)# has zeros #+-i# and #+-sqrt(2)i#

We will use the difference of squares identity, which can be written:

#a^2-b^2 = (a-b)(a+b)#

with #a=x# and #b=i# or #b=sqrt(2)i# as follows:

#f(x) = 5x^4+15x^2+10#

#color(white)(f(x)) = 5(x^4+3x^2+2)#

#color(white)(f(x)) = 5(x^2+1)(x^2+2)#

#color(white)(f(x)) = 5(x^2-i^2)(x^2-(sqrt(2)i)^2)#

#color(white)(f(x)) = 5(x-i)(x+i)(x-sqrt(2)i)(x+sqrt(2)i)#

Hence the zeros of #f(x)# are:

#x = +-i#

#x = +-sqrt(2)i#

Serenity Johnson · Answer

undefined To find all zeros of the function $f(x) = 5x^4 + 15x^2 + 10$, you can follow these steps:

1. Set $f(x)$ equal to zero:

$$5x^4 + 15x^2 + 10 = 0$$

2. Factor out the common factor, if possible. In this case, there is no common factor among the terms.

3. Make a substitution to simplify the equation. Let $y = x^2$, then the equation becomes:

$$5y^2 + 15y + 10 = 0$$

4. Solve the quadratic equation $5y^2 + 15y + 10 = 0$ using the quadratic formula or factoring.

5. Once you find the solutions for $y$, substitute them back into $y = x^2$ to find the corresponding values of $x$.

6. These values of $x$ are the zeros of the function $f(x)$.