# Given #{(p(x)=x^4+a x^3+b x^2+c x+1),(q(x)=x^4+c x^3+b x^2+a x + 1):}# find the conditions for #a, b, c, (a ne c)# such that #p(x)# and #q(x)# have two common roots, then solve #p(x)=0# and #q(x) = 0#?

##### 2 Answers

The zeros are

#### Explanation:

Given:

#p(x) = x^4+ax^3+bx^2+cx+1#

#q(x) = x^4+cx^3+bx^2+ax+1#

with two common roots and

Note that

If

#0 = p(x_1) - q(x_1)#

#color(white)(0) = (a-c)x_1^3+(c-a)x_1#

#color(white)(0)= (a-c)x_1(x_1-1)(x_1+1)#

Hence the two roots are

Then:

#0 = p(1) = a+b+c+2#

#0 = p(-1) = -a+b-c+2#

Adding and subtracting these two equations, we find:

#b = -2#

#a+c = 0#

See bellow.

#### Explanation:

Given

follows that

so

and

now

Equating for all

for

and for

So the polynomials are