One of the roots of the following equation is 1, which is a double root. Find the remaining roots. x4 - 2x² + 1 = 0

Question

One of the roots of the following equation is 1, which is a double root. Find the remaining roots.

x⁴ - 2x² + 1 = 0

Answer 1

the Remainder Theorem and Synthetic Division are both tools that we can use.

With the remainder theorem we replace the variable with a potential root and evaluate the polynomial. If the resulting value = 0 then the value substituted IS a root.

We are given that 1 is a double root so: P(1) = (1)⁴ -2(1)² +1 =0 as it should be so we take it down by one degree P(x) = (x)³ +1(x)² -1(x) -1 and use the second root of x=1 P(1) = (1)³ + (1)² -1(1) -1 = 0 also.

Now we have a second degree polynomial P(x) = x² + 2x +1 which can be factored into (x+1) (x+1) or (x+1)².

setting those two factors equal to zero and solving for x we find that the other two roots are also a double of (-1). So the four roots are x = 1,1,-1,-1. If this were graphed, the curve would "bounce" at both 1 & -1 on the x axis.