document.write( "Question 1162880: f(x) is a polynomial of degree 4 with f(x) + 1 divisible by (x + 1)^2 and f(x) itself divisible by x^3. What is the value of f(−4)? \n" ); document.write( "
Algebra.Com's Answer #786894 by greenestamps(13206)\"\" \"About 
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\n" ); document.write( "f(x) is a polynomial of degree 4 which is divisible by x^3. So

\n" ); document.write( "\"f%28x%29+=+ax%5E4%2Bbx%5E3\"

\n" ); document.write( "f(x)+1 is divisible by (x+1)^2 = x^2+2X+1.

\n" ); document.write( "Perform the polynomial long division....
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document.write( "                              ax^2 + (b-2a)x + (3a-2b)\r\n" );
document.write( "            ----------------------------------------------\r\n" );
document.write( "   x^2+2x+1 ) ax^4 +    bx^3 +       0x^2   +   0x   +   1\r\n" );
document.write( "              ax^4 + (2a)x^3 +       ax^2\r\n" );
document.write( "              -------------------------------\r\n" );
document.write( "                   (b-2a)x^3 -       ax^2 +      0x\r\n" );
document.write( "                   (b-2a)x^3 + (2b-4a)x^2 + (b-2a)x\r\n" );
document.write( "                   ---------------------------------------\r\n" );
document.write( "                               (3a-2b)x^2 +  (2a-b)x +  1\r\n" );
document.write( "                               (3a-2b)x^2 + (6a-4b)x + (3a-2b)\r\n" );
document.write( "                               ---------------------------------\r\n" );
document.write( "                                            (3b-4a)x + (1-3a+2b)

\n" ); document.write( "The remainder has to be zero, so

\n" ); document.write( "\"3b-4a+=+0\"
\n" ); document.write( "\"1-3a%2B2b+=+0\"

\n" ); document.write( "Solve the pair of equations (I leave that much to you) to find a=3 and b=4.

\n" ); document.write( "So the function is

\n" ); document.write( "\"f%28x%29+=+3x%5E4%2B4x%5E3\"

\n" ); document.write( "ANSWER:
\n" ); document.write( "\"f%28-4%29+=+3%28-4%29%5E4%2B4%28-4%29%5E3+=+3%28256%29%2B4%28-64%29+=+768-256+=+512\"
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