Question

What are all of the real roots of the following polynomial? f(x) = x^4 -24x^2 - 25 A. -1 or 1 B. -5 or 5 C. -5, -1, 1, or 5 D. -5 or -1 ?

Answer 1

For this case we have the following function:

$f (x) = x ^ 4-24x ^ 2-25$

We make a change of variable:

$u = x ^ 2$

So, we have:

$u ^ 2-24u-25$

We apply the resolver:

$a = 1\\b = -24\\c = -25$

$u = \frac {-b \pm \sqrt {b ^ 2-4 (a) (c)}} {2 (a)}\\u = \frac {- (- 24) \pm \sqrt {(- 24) ^ 2-4 (1) (- 25)}} {2 (1)}\\u = \frac {24 \pm \sqrt {576 + 100}} {2}\\u = \frac {24 \pm \sqrt {676}} {2}\\u = \frac {24\pm26} {2}$

We have two roots:

$u_ {1} = \frac {24 + 26} {2} = 25\\u_ {2} = \frac {24-26} {2} = - 1$

Returning the change:

$x ^ 2 = 25\\x = \pm \sqrt {25} = \pm5$

$x ^ 2 = -1$ Give the imaginary roots.

Answer:

Real roots $\pm5$