SOLUTION: 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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Question 1195206: If x+2 and x-3 are factors of the polynomialp%28x%29=x%5E3%2B5x%5E2%2Bax%2Bb, find a.
Found 5 solutions by lotusjayden, ikleyn, Theo, Solver92311, greenestamps:
Answer by lotusjayden(18) About Me  (Show Source):
You can put this solution on YOUR website!
I'll solve this part by myself.
(b)   If x%2B2 and x-3 are factors of the polynomial p%28x%29+=+x%5E3%2B5x%5E2%2Bax%2Bb,
        find a.
~~~~~~~~~~~~~~~~~~ 
First, you should multiply the two known factors together: 

%28x%2B2%29%28x-3%29

which simplifies to:
x%5E2-x-6

after you get that, one method to solve this problem is to do trial and error:

After a while, you should get:

%28x%5E2-x-6%29%28x%2B6%29

The simplified version is:
x%5E3%2B5x%5E2-12x-36

Now, you can see that a=-12, and b=-36.

Thus we get the ANSWER :  a = -12.

   Solved.




(hehe solved my own problem)

Answer by ikleyn(52776) About Me  (Show Source):
You can put this solution on YOUR website!
.
If x+2 and x-3 are factors of the polynomial p%28x%29=x%5E3%2B5x%5E2%2Bax%2Bb, find a.
~~~~~~~~~~~~~~~~~~~~~~~~~ 

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.


ANSWER.  a = -12.

Solved.



Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
i get A = -12 and B = -36
checking with those values makes the remainder 0, confirming those values are good.
i used synthetic division to solve this.
here's my worksheet.

here's a reference on how to do synthetic division.
https://www.purplemath.com/modules/synthdiv.htm

Answer by Solver92311(821) About Me  (Show Source):
You can put this solution on YOUR website!


If is a factor of the polynomial, then is a zero of the polynomial. Likewise, is a zero of the polynomial.

Using Synthetic Division:

 3  1  5    a        b
       3   24        a + 24
   -----------------------------
    1  8    a + 24  3a + 72 + b

So
Also:
-2  1  8   a + 24
      -2     - 12
   ------------------
    1  6   a + 12

So

John

My calculator said it, I believe it, that settles it

From
I > Ø

Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


You have already received four responses, all showing different ways to solve the problem.

Here is yet another way that might teach you something useful.

In a cubic equation with leading coefficient 1,

x%5E3%2Bbx%5E2%2Bcx%2Bd=0

with roots p, q, and r, the following are the relationships between the roots and the coefficients of the equation:

(1) b = -(p+q+r)
(2) c = (pq+pr+qr)
(3) d = -(pqr)

Note it is easy to see these relationships by writing the polynomial equation in factored form and expanding:

%28x-p%29%28x-q%29%28x-r%29=x%5E3%2B%28-p-q-r%29x%5E2%2B%28pq%2Bpr%2Bqr%29x-pqr

To use these facts to solve this problem....

Since x+2 and x-3 are factors of the polynomial, two of the roots are p=-2 and q=3.

The coefficient of the x^2 term is 5, so from (1) above
-%28p%2Bq%2Br%29=5
-%28%28-2%29%2B%283%29%2Br%29=5
-%281%2Br%29=5
1%2Br=-5
r=-6

So the third root is r=-6.

The problem asks for the coefficient of the x (linear) term. From (2) above, the coefficient of the linear term is

a=pq%2Bpr%2Bqr=%28-2%29%283%29%2B%28-2%29%28-6%29%2B%283%29%28-6%29=-6%2B12-18=-12

ANSWER: a = -12