Fundamental Theorem of Algebra

Question

I need help with these very complex algebra 2 problems. Thank you.

f(x)= x⁴+21x² -100

Answer 1

If you want to factor f(x):

x⁴ + 21x² -100 =(x² + 25)(x² - 4)

using the perfect square difference formula (a² - b²) = (a +b)(a-b)

(x² - 4) = (x + 2)(x - 2)

x⁴ + 21x² -100=(x² + 25)(x - 2)(x + 2)

If you want to find the solution (roots) of f(x):

(x² + 25)(x - 2)(x + 2) = 0

(x² + 25)=0 or (x - 2)= 0 or (x +2) =0

The 4 roots are:

x = 5i, x =-5i, x = 2, and x = -2

i is the imaginary = sqr(-1)

Answer 2

It isn't at all clear what you are trying to find. I will assume that you are trying to find the roots of this equation.

First, make the substitution that y = x². This changes your equation to

g(y) = y² + 21y - 100 This is in a form that you can factor easily. You are looking for factors of 100 that have a difference of 21. 25 * 4 = 100, and 25 - 4 - 21, so 25 and 4 are the factors of 100. Factor g(y):

g(y) = (y + 25) (y - 4) ==> y = -25 and y = 4, but y = x², so this implies that x² = -25 and x² = 4

The second equation gives x = +/- 2; the result from the first depends on whether you want real solutions or can accept imaginary/complex solutions, since there are no real numbers that have a negative square. If imaginary solutions are possible, then x = +/- 5i