Question

if a²-1 is a factor of pa^4+qa³+ra²+sa+t show that p+t+r=q+s=0

Answer 1

pa^4+qa³+ra²+sa+t

a²-1 is a factor of the polynomial.

a² = 1
a = √1
a = ±1

Put a = 1,

pa^4+qa³+ra²+sa+t = 0
p(1)^4 + q(1)³ + r(1)² + s(1) + t = 0
p(1) + q(1) + r(1) + s + t = 0
p + q + r + s + t = 0

Put a = -1,

pa^4+qa³+ra²+sa+t = 0
p(-1)^4 + q(-1)³ + r(-1)² + s(-1) + t = 0
p(1) + q(-1) + r(1) - s + t = 0
p - q + r - s + t = 0
p + r + t = q + s

In first equation,
p + q + r + s + t = 0
q + s + q + s = 0
2(q + s) = 0
q + s = 0

p + r + t = q + s = 0

Hence proved

Hope it helps

Answer 2

Hi,

Please see the attached file!


Thanks