Arithmetic progression and polynomialsSuppose the quadratic polynomial p ( x ) = a x 2 + b x + chas positive coefficients a,b,c such that these are in AP in the given order. If m and n are the integer zeros of the polynomial then what is the value of m + n + m n? I have tried quadratic formula but ain't getting the answer.

Question

Arithmetic progression and polynomialsSuppose the quadratic polynomial  p  (  x  )  =  a      x    2    +  b  x  +  chas positive coefficients a,b,c such that these are in AP in the given order. If m and n are the integer zeros of the polynomial then what is the value of   m  +  n  +  m  n? I have tried quadratic formula but ain&#039;t getting the answer.

Quinten Cervantes · Accepted Answer

Step 1Since m and n are roots of the polynomial we have                    a                  x          2                +        b        x        +        c                            =        a                  (          x          −          m          )                          (          x          −          n          )                                                  =        a                  x          2                −        a        (        m        +        n        )        x        +        a        m        n            Then,  m  +  n  =  −      b    a      and    m  n  =      c    a  So, since a,b and c are in AP we get  m  +  n  +  m  n  =  −      b    a    +      c    a    =            c      −      b        a    =            b      +      b      −      a      −      b        a    =            b      −      a        a    =      b    a    −  1and       b    a    =  −  (  m  +  n  ), then  m  +  n  +  m  n  =  −  (  m  +  n  )  −  1      ⟹      m  n  +  2  (  m  +  n  )  +  4  =  3Step 2So   (  m  +  2  )  (  n  +  2  )  =  3Now, since 3 is a prime we have   (  m  ,  n  )  ∈      {    (    −    1    ,    1    )    ,    (    1    ,    −    1    )    ,    (    −    3    ,    −    5    )    ,    (    −    5    ,    −    3    )    }  . Notice that   (  m  ,  n  )  =  (  −  1  ,  1  ) or   (  m  ,  n  )  =  (  1  ,  −  1  ) implies   b  =  0, but we know that a,b and c are positive. So,   (  m  ,  n  )  =  (  −  3  ,  −  5  ) or   (  −  5  ,  −  3  ), and then                              m          +          n          +          m          n          =          −          8          +          15          =          7

Arithmetic progression and polynomials. Suppose the quadratic polynomial p(x)=ax^2+bx+c has positive coefficients a,b,c such that these are in AP in the given order.

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