SOLUTION: please help me urgently.... 1)Find m and n if (x-2) and (x+3) are both factors of x^3+mx^2-nx-6 2) When p(x)=x^4-3x^3+ax^2+bx-6 is divided by (x+1)it has a remainder of 8. If (x

Question 526662: please help me urgently....
1)Find m and n if (x-2) and (x+3) are both factors of x^3+mx^2-nx-6
2) When p(x)=x^4-3x^3+ax^2+bx-6 is divided by (x+1)it has a remainder of 8. If (x-3) is a factor of p(x) find a and b.
3) If x^2-4 is a factor of 2x^3-ax^2+bx+4 find a and b.

Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
When you divide a polynomial P(x) by a divisor polynomial D(x) you get a quotient polynomial Q(x) and a remainder polynomial R(x). With some luck, all of those polynomial will be very small and simple, and maybe the remainder will be 0, or just a number (a polynomial of degree zero).
1) (remainder is zero) means that

or
Similarly
leads you to
Those two equations form a linear system that allows you to find m and n.
2) Same idea
(remainder is zero) so and
(remainder is 8) so
You'll get two equations that would allow you to find a and b.
3) (remainder is zero)
I would bet on
and
Multiplying will tell you if that product is a polynomial that could be the one you were given. (Easy way to solve it).
You are probably expected to use
and
to set up a system of equations to solve for a and b, but who really enjoys solving systems of equations?
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