Given the polynomial f(x) = 4x^3 - 12x^2 + ax + b, find a,b, if f(x) is divisible by (x^2 - 1).

Given f(x) = 4x^3 - 12x^2 + ax +
b

Given that f(x) is divided by
(x^2-1)

Then (x^2 -1) is a factor of
f(x).

Then the roots of (x^2 -1) are the solutions to the
function f(x).

==> x^2 -1 =
0

==> x1 = 1

==>
x2= -1

Then x = -1 and x= 1 are roots of
f(x).

==> f(1) = f(-1) =
0

Let us
substitute.

==> f(1) = 4(1^3) - 12(1^2) + a(1) + b =
0

==> 4 - 12 + a + b =
0

==> a + b = 8
.............(1)

==> f(-1) = 4(-1)^3 -12(-1^2) +
a(-1) + b = 0

==> -4 - 12 - a + b =
0

==> -a + b = 16
...............(2).

Now we will add (1) and
(2).

==> 2b =
24

==> b =
12

==> a =
-4

==> f(x) = 4x^3 -
12x^2 -4x + 12