SOLUTION: The expression x^3 + ax^2 + bx + 3 is exactly divisible by x+3 but it leaves a remainder of 91 when divided by x-4. What is the remainder when it is divided by x+2?

Algebra.Com
Question 1183627: The expression x^3 + ax^2 + bx + 3 is exactly divisible by x+3 but it leaves a remainder of 91 when divided by x-4. What is the remainder when it is divided by x+2?
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
The expression x^3 + ax^2 + bx + 3 is exactly divisible by x+3
but it leaves a remainder of 91 when divided by x-4.
What is the remainder when it is divided by x+2?
~~~~~~~~~~~~~~~~~


            The solution is to apply the Remainder theorem several times. 


(1)  Due to the Remainder theorem, first condition means that the numner -3 is the root of the given polynomial.

     It gives you this equation

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

     which is equivalent to

         9a - 3b = 24,   or    3a - b = 8.      (1)



(2)  Due to the same theorem, second condition means that the value of the given polynomial is 91 at x= 4.

     It gives you this equation

         4^3 + a*4^2 + b*4 + 3 = 91,

     which is equivalent to

         16a + 4b = 24,   or   4a + b = 6.     (2)



(3)  Adding equations(1) and (2), we get  7a = 14,  a = 14/7 = 2.

     Then from (1),  3*2 - b = 8,  b = 6 - 8 = -2.


     So, the polynomial is  p(x) = x^3 + 2*x^2 - 2*x + 3.



(4)  Due to the Remainder theorem, the remainder of the polynomial when it is divided by x+2 is the value of the polynomial at x= -2, i.e.

         (-2)^3 + 2*(-2)^2 -2*(-2) + 3 = 7.      ANSWER

Solved.



RELATED QUESTIONS

The expression f(x)=x^3+ax^2+bx-6 is exactly divisible by (x-1), and has a remainder of... (answered by reviewermath)
The expression x^4 + ax^3 + 5x^2 + bx + 6 when divide by (x-2), the remainder is 16 and... (answered by solver91311)
when the expression x^4+ax^3+5x^2+bx+6 is divided by(x-2), the remainder is 16. when it... (answered by josgarithmetic)
The expression ax˛+bx+c is divisible by (x-1),has remainder 2 when divided by (x+1) and... (answered by Boreal)
When f(x)= ax^2+bx-4 is divided by x+2, the remainder is -2. When f(x) is divided by x+3, (answered by josgarithmetic)
If ax^3+bx^3+x has (x+2) as a factor and leaves a remainder 4 when divided by (x-2).Find... (answered by solver91311)
If ax^3+bx^2+x-6 has (x+2) as a factor and leaves a remainder 4 when divided by... (answered by solver91311)
If f(x)=ax^4 +bx^3 -x^2 +2x +3 has a remainder 11 when divided by x-2and has a remainder... (answered by josgarithmetic)
The polynomial R(x) = x^4 + Ax^3 + Bx^2 + 10x - 1 has a remainder of -15 when divided by... (answered by josgarithmetic)