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Find all integers $n$ such that the quadratic $7x^2 + nx - 11$ can be expressed as the product of two linear factors with integer coefficients.

 Feb 18, 2020
 #1
avatar+109740 
+2

7x^2  + nx  -  11

 

We  have these possiibilities

 

(7x  - 11) ( x + 1)  →  7x^2 - 11x + 7x - 11  → n = -4

 

(7x  + 1) ( x - 11) →  7x^2 -77x + 1x - 11 →  n = -76

 

(7x + 11) ( x - 1)   →  7x^2  + 11x - 7x - 11 →  n  = 4

 

(7x - 1) ( x + 11) →  7x^2 + 77x - 1x - 11 →  n = 76

 

 

 

cool cool cool

 Feb 18, 2020
 #2
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+1

Thanks! But what if it were (-7x-_)(-x+_) as well? Would the polynomials follow the same pattern?

Guest Feb 18, 2020
 #3
avatar+109740 
+1

Yeah  because  notice  that

 

(-7x -  ___)  (-x + ___)    is really the same as

 

(-1) *(7x +  ___)  * ( -1) *  (x - ___)   =

 

(-1) (-1) ( 7x +  ___) ( x - ___)  =

 

(1) (7x + ____)  ( x -  ___) =

 

(7x + ___ )  ( x  - ___)

 

 

cool cool cool

CPhill  Feb 18, 2020
 #4
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+1

Ah, thank you!

Guest Feb 18, 2020

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