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

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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 **

We have to find a,b, if f=4x^3-12x^2+ax+b is divisible by (x^2 - 1)

Now x^2 - 1 = (x + 1)(x - 1)

If f = 4x^3 - 12x^2 + ax + b is divisible by (x^2 - 1) it is divisible by both (x + 1) and (x - 1).

Now we use the remainder theorem. The divisibility gives us that:

f( 1) = 0

=> 4x^3 - 12x^2 + ax + b =0

=> 4 - 12 + a + b = 0

=> a + b = 8

f(-1) = 0

=> 4x^3 - 12x^2 + ax + b =0

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

=> -a + b = 16

Add a + b = 8 and -a + b = 16

=> 2b = 24

=> b = 12

a = 12 - 16

=> a = - 4

**Therefore a = -4 and b = 12.**

If the polynomial f is divided by x^2 - 1, then the roots of the polynomial x^2 - 1 are the roots of f.

We'll determne the roots of x^2 - 1:

x^2 - 1 = 0

x^2 - 1 = (x-1)(x+1)

We'll put x - 1 = 0.

x = 1

We'll put the next factor, x + 1 = 0

x = -1

We'll substitute x from f by 1 and -1 and we'll get:

f(1) = 4 - 12 + a + b

f(1) = 0

We'll combine like terms:

a + b - 8 = 0

a + b = 8 (1)

f(-1) = -4 - 12 - a + b

f(-1) = 0

- a + b = 16 (2)

We'll add (1) + (2):

a + b - a + b = 8 + 16

We'll combine and eliminate like terms:

2b = 24

**b = 12**

We'll substitute b in (1):

a + b = 8

a + 12 = 8

a = 8 - 12

**a = -4**

**The polynomial f is: f = 4x^3 - 12x^2 - 4x + 12**