Question 1075892
.
A polynomial has a remainder of -6 and 4 when divided by (x+1) and (x-1) respectively. 
Find the remainder when the polynomial is divided by x^2-1.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


<pre>
The fact that "A polynomial has a remainder of -6 when divided by (x+1)" means P(-1) = -6.   (The "Remainder theorem")


The fact that "A polynomial has a remainder of 4 when divided by (x-1)" means P(1) = 4.   (The same theorem)


Now, looking for the remainder when the polynomial is divided by x^2-1, we can write for this remainder


P(x) = Q(x)*(x^2-1) + ax + b   (*)    ( after division P(x) by (x^2-1) )


If you put x= -1 in this equation (*), you will get

P(-1) = -6 = a*(-1) + b.


If you put x= 1 in this equation (*), you will get

P(1) = 4  = a*1 + b.


Thus you have these two equations

-a + b = -6,
 a + b =  4.


Solve it by any method you know to get b = -1, a = 5.


<U>Answer</U>.  The remainder under the question is  (5x -1).
</pre>

Solved.



On the "Remainder theorem" see the lesson

&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</A>

in this site.



&nbsp;&nbsp;&nbsp;&nbsp; <B>Theorem</B> &nbsp;&nbsp;(the <B><I>remainder theorem</I></B>)


<B>1</B>. The remainder of division the polynomial &nbsp;{{{f(x)}}}&nbsp; by the binomial &nbsp;{{{x-a}}}&nbsp; is equal to the value &nbsp;{{{f(a)}}}&nbsp; of the polynomial. 

<B>2</B>. The binomial &nbsp;{{{x-a}}}&nbsp; divides the polynomial &nbsp;{{{f(x)}}}&nbsp; if and only if the value of &nbsp;{{{a}}}&nbsp; is the root of the polynomial &nbsp;{{{f(x)}}}, &nbsp;i.e. &nbsp;{{{f(a) = 0}}}.

<B>3</B>. The binomial &nbsp;{{{x-a}}}&nbsp; factors the polynomial &nbsp;{{{f(x)}}}&nbsp; if and only if the value of &nbsp;{{{a}}}&nbsp; is the root of the polynomial &nbsp;{{{f(x)}}}, &nbsp;i.e. &nbsp;{{{f(a) = 0}}}.



Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic 
"<U>Divisibility of polynomial f(x) by binomial (x-a). The Remainder theorem</U>".