document.write( "Question 1154812: a. Find a polynomial of minimum degree such that when divided by x+2 has a remainder of -1 and when divided by x-1 has a remainder of 3.
\n" ); document.write( "b. Find a polynomial of degree 3 such that when divided by x^2-5x has a remainder of 6x-15 and when divided by x^2-5x+8 has a remainder of 2x-7. \n" ); document.write( "

Algebra.Com's Answer #777295 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " I will solve part (a) ONLY.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "It is clear that the polynomial can not be linear (of the degree 1).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, I will find such a polynomial of the degree 2 (quadratic).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Let f(x) = x^2 + bx + c be such a polynomial.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "According to the Remainder theorem, the imposed conditions are equivalent to \r\n" );
document.write( "\r\n" );
document.write( "    f(-2) = -1  and  f(1) = 3,   or\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    (-2)^2 - 2b + c = -1      (1)\r\n" );
document.write( "\r\n" );
document.write( "    1^2    +  b + c =  3      (2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Equations (1) and (2) are equivalent to\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "           - 2b + c = -5      (3)\r\n" );
document.write( "\r\n" );
document.write( "              b + c =  2      (4)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From equation (3), subtract equation (4). You will get\r\n" );
document.write( "\r\n" );
document.write( "            -3b     = -7;   hence,  b = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then from (4),  c = 2 - b = 2 -  = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, the polynomial is  f(x) =  .    ANSWER\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "CHECK.  f(-2) = (-2)^2+(7/3)*(-2) - 1/3 = 4 - 14/3 - 1/3 = -1;\r\n" );
document.write( "\r\n" );
document.write( "        f(1) = 1^2 + 7/3 - 1/3 = 1 + 2 = 3.    ! Correct !\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\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" );