Question 1183420
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Pls help me get remainder when 2x^4 - 5x^3 + 7x^2 - x + 6 is divided by x^2+x-2 by using only remainder theorem. 
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            @MathLover1 solved the problem, but used the long division of polynomials

            instead of the Remainder theorem, as it was requested in the post.


            Below I developed the solution to this problem using the Remainder theorem - - - as requested.           E N J O Y  ( ! )



<pre>
Since the divisor  (x^2 + x - 2)  is the quadratic polynomial, the reminder of the division is the linear binomial
and, therefore, has the form  (ax+b).

Our task is to find / (to determine) the coefficients "a" and "b" of this linear binomial.


So, we have

    2x^4 - 5x^3 + 7x^2 - x + 6 = q(x)*(x^2+x-2) + (ax+b).     (1)


where q(x) is another quadratic polynomial, but I even will not touch it . . . 

 
Now notice that the roots of the quadratic polynomial (x^2+x-2) are -2 and 1
(because  x^2+x-2 = (x+2)*(x-1) in the factored form).



The remainder of division the quartic polynomial  2x^4 - 5x^3 + 7x^2 - x + 6  by  (x+2)  is the value of this quartic 
at x= -2 (the Remainder theorem), i.e.

    2*((-2)^4) - 5*((-2)^3) + 7*((-2)^2) - (-2) + 6 =  108.


Due to equality (1), it means that the linear binomial (ax+b)   has the value of  108  at  x= -2,  i.e.

    -2a + b = 108     (2)



The remainder of division the quartic polynomial  2x^4 - 5x^3 + 7x^2 - x + 6  by  (x-1)  is the value of this quartic 
at x= 1 (the Remainder theorem), i.e.

    2*(1^4) - 5*(1^3) + 7*(1^2) - 1 + 6 =  9.


Due to equality (1), it means that the linear binomial (ax+b)   has the value of 9 at x= 1, i.e.

    a + b = 9         (3)


   +-------------------------------------------------------------------------+
   | Thus we have two equations (2) and (3) to find two unknowns "a" and "b. |
   +-------------------------------------------------------------------------+


To solve the system, subtract equation (3) from equation (2) to get

    -2a - a = 108-9,   -3a = 99,  a = 99/(-3) = -33.


Then from (3) you get  b= 9 - a = 9 - (-33) = 9 + 33 = 42.


Thus the problem is just solved (by the method you requested), and the <U>ANSWER</U> is:

    +---------------------------------------------------------------+
    |    the remainder of division of  2x^4 - 5x^3 + 7x^2 - x + 6   |
    |              by  (x^2+x-2)  is  -33x + 42.                    |
    +---------------------------------------------------------------+
</pre>

Solved as requested and carefully explained.