for what value of a,the polynomial g(x)is a factor of f(x)

Question

f(x)=2 x*x*x x+3*x*x*x +2 a*x*x+3 x+6,g(x)=x+2

Answer 1

Another process that can also be used is synthetic division.  You divide the coefficients of f(x) by the root of f(x).  Since g(x) is your factor, your root would be -2.   This is also the divisor.  The quotient must have a remainder of zero.

 

 

-2 | 2      3           2a                3                     6

 

             -4            2            (-4a-4)             (8a+2)

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      2     -1         (2a+2)        (-4a-1)               0

 

 

 

As you can see, the last digit under the line is zero.  This means that (x + 2) is a factor of f(x).  Now we set the equation to solve for a.  Use the last column in the division process to create it.

 

6 + 8a + 2 = 0

 

8a = -8

 

  a = -1

Answer 2

f(x) = 2x⁴+3x³+2ax²+3x+6

 

By the Factor Theorem, x+2 is a factor of f(x) if f(-2) = 0

 

So, we must find a so that f(-2) = 0:

 

   f(-2) = 32 - 24 +8a - 6 + 6 = 0

 

                                   8a = -8

 

                                     a = -1