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Consider the quadratic expression \(13x^2 + nx - 17\). For certain values of n, it may be factored into a product of two linear polynomials, both of which have integer coefficients. What are all such values of n?

 Nov 25, 2022
 #1
avatar+1312 
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13 is prime, so we know we can write it as (13x _ _)(x _ _)

17 is also prime so we can guess put it into either one as (13x - 17)(x + 1) or (13x + 1)(x - 17) or (13x - 1)(x + 17) or (13x + 17)(x - 1).

FOILing it out:

(13x - 17)(x + 1) = 13x^2 - 4x - 17

(13x + 1)(x - 17) = 13x^2 - 220x - 17

(13x - 1)(x + 17) = 13x^2 + 220x - 17

(13x + 17)(x - 1) = 13x^2 + 4x - 17

Values of n: -220, -4, 4, 220

 Nov 25, 2022

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