We are given that P(x) has factors (x-1), (x+1) and (x-3). We need to find which of x^2 +1 , x^2-1 and x^2- 4x +3 is also a factor of P(x).
Now, (x+1)*(x-1) gives us (x^2 -1). Therefore as (x-1) and (x+1)
are factors of P(x), (x^2 -1) is also a factor of P(x).
Similarly (x-1)*(x-3) = x^2- 4x +3. Therefore as (x-1) and (x-3) are factors of P(x), x^2- 4x +3 is also a factor of P(x).
Therefore we determine that x^2- 4x +3 and x^2 - 1 are factors of P(x).
We know that we can write a polynomial as a product of linear factors (when we know it's roots).
So, if x - 1, x - 3 and x + 1 are the factors of the polynomial, then we can write P(x) as:
P(x) = (x - 1)(x - 3)(x + 1)
We'll calculate the product between the first 2 factors:
(x-1)(x-3) = x^2 - 3x - x + 3
We'll combine like terms and we'll get:
x^2 - 4x + 3
So one factor of P(x) could be x^2 - 4x+ 3.
We also notice that if we'll multiply (x - 1)(x + 1) = x^2 - 1
So, the right answer is represented by the points b) and c).
Since (x-1), (x-3) and (x+1) are the factors of a polynomial P(x)of degree 3.
Therefore P(x) has the following factors:
(x-1)(x-3) = x^2-4x+3
(x-1)(x+1) = x^2-1.
(x-3)(x+1) = x^2 -2x-3.
Among the given choices the expression x^2+1 is not a factor of P(x).
The choices s at b) x^2-1 and at c) x^2-4x+3 are the factors of P(x).