Question 1200513
.
When the polynomial F(x) of degree 3 is divided by x-1, the remainder is -1.
When F(x) is divided by x+1, the remainder is -5.
If F(x) is divided by (x-1)(x+1), the remainder is ax+b, where a and b are constants .
Find the value of a and b.
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                    Step by step solution.



<pre>
(a)  When the polynomial F(x) of degree 3 is divided by x-1, the remainder is -1.

     According to the Remainder theorem, it means that F(1) = -1.    (1)


(b)  When F(x) is divided by x+1, the remainder is -5.

     According to the Remainder theorem, it means that F(-1) = -5.   (2)



(c)  If F(x) is divided by (x-1)(x+1), the remainder is ax+b, where a and b are constants.

     It means that  

              F(x) = Q(x)*(x-1)*(x+1) + (ax+b).                        (3)



     In this equation (3), put x= 1.  You will get

              F(1) = Q(x)*(1-1)*(1+1) + (a*1+b),  or

              F(1) =                     a + b.

      But, according to (1),  F(1) = -1.   Hence

              a + b = -1.    (4)



      Next, in equation (3), put x= -1.  You will get

              F(-1) = Q(-1)*(1-(-1))*(1+(-1)) + (a*(-1)+b),  or

              F(-1) =                            -a + b.

      But, according to (2),  F(-1) = -5.   Hence

              -a + b = -5.    (5)



(d)   Thus we have this system of two equations for "a" and "b"

              a + b = -1      (4)
             -a + b = -5      (5)


      To solve it, add equations (4) and (5). You will get

                 2b = -1 + (-5) = -6  ===>  b = -6/2 = -3.

      Then from (4)

             a = -1 - b = -1 -1 - (-3) = -1 + 3 = 2.


<U>ANSWER</U>.  a = 2;  b = -1.
</pre>

Solved.


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On the Remainder theorem and many associated solved problems, &nbsp;see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Divisibility-of-polynomial-f%28x%29-by-binomial-x-a.lesson>Divisibility of polynomial f(x) by binomial (x-a) and the Remainder theorem</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Solved-problems-on-the-Remainder-theorem.lesson>Typical problems on the Remainder theorem</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Advanced-problems-on-the-Remainder-theorem.lesson>Advanced problems on the Remainder theorem</A> 

in this site.