Question 1172550
.
The polynomial f(x)=x^3-x^2-6kx+4k^2 where k is a constant has (x-3) as a factor. 

(a) Find the possible values of k and 

(b) for the {{{highlight(cross(integral))}}} <U>integer</U> value of k find the remainder when f(x) is divided by x+2.
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<pre>
According to the Remainder theorem, the fact that the polynomial f(x) = x^3 - x^2 - 6kx + 4k^2 has (x-3) as a factor

means that the value of x= 3 is the root of the polynomial.



It gives this equation for k


    f(3) = 0 = 3^3 - 3^2 - 6*3*k + 4k^2,   or

           4k^2 - 18k + 18 = 0,            which is equivalent to

           2k^2 -  9k +  9 = 0.


The roots of the equation are (use the quadratic formula)  k= 4  and  k= {{{3/2}}}.


Of these two roots, the integer value for k is 4 (four).


At k = 4, the polynomial takes the form  f(x) = x^3 - x^2 - 6*4x + 4*4^2 = x^3 - x^2 - 24x + 64.


The reminder of this polynomial, when divided by (x+2),  it its value at x= -2  (here I apply the Remainder theorem again)


    f(-2) = (-2)^3 - (-2)^2 - 24*(-2) + 64 = 100.


<U>ANSWER</U>.  (a)  the possible values of k are  k= 4  and  k= {{{3/2}}}.

         (b)  for the integer value of k, the remainder when f(x) is divided by x+2 is equal to 100.
</pre>

Solved.  &nbsp;&nbsp;&nbsp;&nbsp;//  &nbsp;&nbsp;&nbsp;&nbsp;All questions are answered.



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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.