Question 1136794
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<pre>
Substitute the value of x = -1 into the polynomial.


The value of the polynomial then will give you the remainder


    3*(-1)^100 + 5*(-1)^95 - 4*(-1)^38 + 2*(-1)^17 = 3*1 + 5*(-1) - 4*1 + 2*(-1) = 3 - 5 - 4 - 2 = -8.     <U>ANSWER</U>
</pre>


It is based on the <U>Remainder Theorem</U>.



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

&nbsp;&nbsp; <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. 

&nbsp;&nbsp; <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}}}.

&nbsp;&nbsp; <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}}}.



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>Solved problems on the Remainder thoerem</A>

in this site.



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

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - 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>".


Save the link to this online textbook together with its description


Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson


to your archive and use it when it is needed.



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