SOLUTION: The remainder of x^3+ax+b when divided by (x^2-1) is x+2. Find a and b.

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: The remainder of x^3+ax+b when divided by (x^2-1) is x+2. Find a and b.      Log On


   

Question 1127209: The remainder of x^3+ax+b when divided by (x^2-1) is x+2. Find a and b.
Found 3 solutions by MathLover1, josgarithmetic, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

if x%5E3%2Bax%2Bb when divided by %28x%5E2-1%29 is x%2B2, then
when %28x%5E2-1%29 multiplied+by++%7B%7B%7Bx%2B2 is x%5E3%2Bax%2Bb
so, multiply %28x%5E2-1%29%28x%2B2%29
x%5E3-x-x-2
x%5E3-2x-2 now compare to x%5E3%2Bax%2Bb and you see that a=-2 and b=-2

Answer by josgarithmetic(39617) About Me  (Show Source):
Answer by ikleyn(52786) About Me  (Show Source):
You can put this solution on YOUR website!
.

            The solution by the tutor @MathLover1 not only is INCORRECT - it is IRRELEVANT, because it operates

            with WRONG CONCEPTIONS.   Unfortunately.   Below find the correct solution. 


From the condition, it should be clear to you that the quotient of division the 3-rd degree polynomial  x^3 + ax + b  
by  the quadratic polynomial  (x^2-1)  is a linear polynomial of the form (x+c) with the leading coefficient 1 
and the constant term c, now unknown to us.  So

    x^3 + ax + b = (x^2-1)*(x+c) + (x+2).


Now make FOIL, combine like terms and then compare coefficients:

    x^3 + ax + b = x^3 -x + cx^2 - c + x + 2 = x^3 + cx^2 + 2-c.


Comparing coefficients, you see that  c= 0  (look in x^2 !);  a= 0  and  b= 2.


Answer.  a= 0  and b= 2.

Solved.

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I estime the tutor @MathLover1 very high - she rarely makes mistakes.

But even first class tutor is not guaranteed of making mistakes - sometimes :)