If x - 1, x - 3 and x + 1 are all factors of a polynomial P(x) of degree 3, which of the following must also be a factor of P(x)?a) x^2 + 1 b) x^2- 1 c) x^2 - 4x + 3

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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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).

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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).

neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

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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).

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