document.write( "Question 1172550: The polynomial f(x)=x^3-x^2-6kx+4k^2 where k is a constant has (x-3)as a factor. Find the possible values of k and for the integral value of k find the remainder when f(x) is divided by x+2. \n" ); document.write( "

Algebra.Com's Answer #797620 by ikleyn(52790) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "The polynomial f(x)=x^3-x^2-6kx+4k^2 where k is a constant has (x-3) as a factor. \r
\n" ); document.write( "\n" ); document.write( "(a) Find the possible values of k and \r
\n" ); document.write( "\n" ); document.write( "(b) for the $\"highlight%28cross%28integral%29%29\"$ integer value of k find the remainder when f(x) is divided by x+2.
\n" ); document.write( "~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "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\r\n" );
document.write( "\r\n" );
document.write( "means that the value of x= 3 is the root of the polynomial.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It gives this equation for k\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    f(3) = 0 = 3^3 - 3^2 - 6*3*k + 4k^2,   or\r\n" );
document.write( "\r\n" );
document.write( "           4k^2 - 18k + 18 = 0,            which is equivalent to\r\n" );
document.write( "\r\n" );
document.write( "           2k^2 -  9k +  9 = 0.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The roots of the equation are (use the quadratic formula)  k= 4  and  k= .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Of these two roots, the integer value for k is 4 (four).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "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.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The reminder of this polynomial, when divided by (x+2),  it its value at x= -2  (here I apply the Remainder theorem again)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    f(-2) = (-2)^3 - (-2)^2 - 24*(-2) + 64 = 100.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER.  (a)  the possible values of k are  k= 4  and  k= .\r\n" );
document.write( "\r\n" );
document.write( "         (b)  for the integer value of k, the remainder when f(x) is divided by x+2 is equal to 100.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.     //     All questions are answered.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "------------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "   Theorem   (the remainder theorem)\r
\n" ); document.write( "\n" ); document.write( "   1. The remainder of division the polynomial $\"f%28x%29\"$ by the binomial $\"x-a\"$ is equal to the value $\"f%28a%29\"$ of the polynomial. \r
\n" ); document.write( "\n" ); document.write( "   2. The binomial $\"x-a\"$ divides the polynomial $\"f%28x%29\"$ if and only if the value of $\"a\"$ is the root of the polynomial $\"f%28x%29\"$ , i.e. $\"f%28a%29+=+0\"$ .\r
\n" ); document.write( "\n" ); document.write( "   3. The binomial $\"x-a\"$ factors the polynomial $\"f%28x%29\"$ if and only if the value of $\"a\"$ is the root of the polynomial $\"f%28x%29\"$ , i.e. $\"f%28a%29+=+0\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "See the lessons\r
\n" ); document.write( "\n" ); document.write( "    - Divisibility of polynomial f(x) by binomial x-a and the Remainder theorem\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on the Remainder thoerem\r
\n" ); document.write( "\n" ); document.write( "in this site.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic
\n" ); document.write( "\"Divisibility of polynomial f(x) by binomial (x-a). The Remainder theorem\".\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Save the link to this online textbook together with its description\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-I
\n" ); document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "to your archive and use it when it is needed.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );