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