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

Answer:Step-by-step explanation:We will make a substitution to make our work easier (when we get there).  We also need to know that[tex]i^2=-1[/tex]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[tex]x^4=u^2[/tex]Therefore,[tex]x^2=u[/tex]Now we will write the polynomial in terms of u instead of x:[tex]f(x)=u^2+21u-100[/tex]Solve for the values of u by setting the polynomial equal to 0 and factoring.  When you factor, you will get:[tex](u+25)(u-4)[/tex]But don't forget that[tex]x^2=u[/tex]so we have to put those back in now:[tex](x^2+25)(x^2-4)=0[/tex]By the Zero Product Property, either[tex]x^2+25=0[/tex] or[tex]x^2-4=0[/tex]We will factor the first term.  Solving for x-squared gives us:[tex]x^2=-25[/tex] andx = ±√-25which simplifies down tox = ±√-1 × 25we can sub in an i-squared for the -1:x = ±√[tex]i^2*25[/tex]The square root of i-squared is "i" and the square root of 25 is 5, sox = ±5iThe next one is a bit easier.  If[tex]x^2=4[/tex], thenx = ±2You 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