Question 1154812
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            I will solve part  (a)  ONLY.



<pre>
It is clear that the polynomial can not be linear (of the degree 1).


So, I will find such a polynomial of the degree 2 (quadratic).


Let f(x) = x^2 + bx + c be such a polynomial.


According to the Remainder theorem, the imposed conditions are equivalent to 

    f(-2) = -1  and  f(1) = 3,   or


    (-2)^2 - 2b + c = -1      (1)

    1^2    +  b + c =  3      (2)


Equations (1) and (2) are equivalent to


           - 2b + c = -5      (3)

              b + c =  2      (4)


From equation (3), subtract equation (4). You will get

            -3b     = -7;   hence,  b = {{{7/3}}}.


Then from (4),  c = 2 - b = 2 - {{{7/3}}} = {{{-1/3}}}.


So, the polynomial is  f(x) =  {{{x^2 + (7/3)x - 1/3}}}.    <U>ANSWER</U>


<U>CHECK</U>.  f(-2) = (-2)^2+(7/3)*(-2) - 1/3 = 4 - 14/3 - 1/3 = -1;

        f(1) = 1^2 + 7/3 - 1/3 = 1 + 2 = 3.    ! Correct !
</pre>

Solved.


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&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.