Question

please guys help me in question 32 if you can .....

Answer 1

x²-1 = (x-1) (x+1) is a factor of f(x) = ax³ + bx² +cx +d = 0

so f(1) = a + b + c + d = 0      ..............(1)
and f(-1) = -a + b - c+ d = 0           ...........(2)

subtracting  2 from 1 gives

2a + 2c = 0  ⇒ a +c = 0

Hence proved.

Answer 2

x²-1=0
x²=1
x=+1,-1
ax³+bx²+cx+d = a(1)³+b(1)²+ c(1) +d = a+b+c+d = 0.....(1)
a(-1)³+b(-1)²+c(-1)+d = -a+b-c+d = 0....(2)
From eqn. (1) and (2)
a+b+c+d = -a+b-c+d
a+a+c+c = 0
2(a+c) = 0
a+c = 0