Determine how many, what type, and find the roots for f(x) = x4  + 21x2  − 100.

Step-by-step explanation:

We will make a substitution to make our work easier (when we get there).  We also need to know that

We will use that as another substitution.  First, let's make the job of factoring a bit easier.  Here's the first substitution.  We will let

Therefore,

Now we will write the polynomial in terms of u instead of x:

Solve for the values of u by setting the polynomial equal to 0 and factoring.  When you factor, you will get:

But don't forget that

so we have to put those back in now:

By the Zero Product Property, either

or

We will factor the first term.  Solving for x-squared gives us:

and

x = ±√-25

which simplifies down to

x = ±√-1 × 25

we can sub in an i-squared for the -1:

x = ±√

The square root of i-squared is "i" and the square root of 25 is 5, so

x = ±5i

The next one is a bit easier.  If

, then

x = ±2

You can see you have 4 solutions.  But you knew that already, since this is a 4th degree polynomial.  The types of solutions are:  2 real, 2 imaginary

c. y = 1/4 x + 5

step-by-step explanation:

first, put the equation of the line on the graph into slope-intercept form

(y = mx + b):   y = -4x + 8

second, a perpendicular line has to have "an opposite/reciprocal slope" to the original slope. so, find the "m" value in your original equation (it's -4, right? )

original slope: m = -4   perpendicular slope: change the sign to its opposite and find the reciprocal ("flip" the number)

new slope is now positive (b/c the original slope was negative 4), and it's 1/4 (not the original slope of 4 or 4/1)