Question 1195206
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If  x+2  and  x-3  are factors of the polynomial  {{{p(x)=x^3+5x^2+ax+b}}}, find a.
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<pre>
If (x+2) and (x-3) are factors of the polynomial  p(x)=x^3+5x^2+ax+b, it means,

due to the Remainder theorem, that x= -2 and x= 3 are the roots of the polynomial.


So, we substitute x= -2 and x= 3 into the polynomial and equate it to zero.

It gives us two equation for two unknowns "a" and "b"


    (-2)^3 + 5*(-2)^2 - 2a + b = 0    (1)

       3^3 + 5*3^2    + 3a + b = 0    (2)


Collect like terms and simplify (1) and (2)

     12 - 2a + b = 0    (3)

     72 + 3a + b = 0    (4)


Now subtract equation (4) from equation (3).  You will get

     60 + 5a     = 0,


which implies  5a = -60,  a = -60/5 = -12.


<U>ANSWER</U>.  a = -12.
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Solved.