Question

In this exercise, we see that sometimes two numbers written using radicals may be equal to each other, without this equality being obvious. Let p(x)=x^4-10x^2+1.

(a) let r_1=sqrt(2) + sqrt(3), r_2= sqrt(2) - sqrt(3), r_3= -sqrt(2) + sqrt(3), r_4=-sqrt(2) - sqrt(3)

(b) Using the quadratic formula, find expressions for the four roots of p(x).

(c) Identify each of the roots in (b) as r_1, r_2,r_3, or r_4. How many strategies can you find?

(d) If you had been given p(x) and had been asked to find all the roots, would you have been more likely to find the list in (a) or in (b)?

(e) Using the roots of p(x), find three different ways of factoring p(x) into a product of two quadratic polynomials. Do you prefer using the list of roots in (a) or in (b)?

Answer 1