document.write( "Question 471434: Find all of the real zeros of the polynomial function, then use the real zeros to factor f over the real numbers.\r
\n" );
document.write( "\n" );
document.write( "f(x)=5x4-7x³+17x²-21x+6\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "Answers\r
\n" );
document.write( "\n" );
document.write( "A. -3, -1, 1, (2/5) ; f(x) = (x - 1)(5x - 2)(x + 1)(x + 3)
\n" );
document.write( "
\n" );
document.write( "B. -3, -1, 1, (-2/5) ; f(x) = (x - 1)(5x + 2)(x + 1)(x + 3)
\n" );
document.write( "
\n" );
document.write( "C. 3, (2/5); f(x) = (x - 3)(5x - 2)(x^2 + 1)
\n" );
document.write( "
\n" );
document.write( "D. 1, (2/5); f(x) = (x - 1)(5x - 2)(x^2 + 3) \r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( " \n" );
document.write( "
Algebra.Com's Answer #323330 by Edwin McCravy(20059)![]() ![]() You can put this solution on YOUR website! \r\n" ); document.write( "\r\n" ); document.write( "f(x)=5x4-7x³+17x²-21x+6\r\n" ); document.write( "\r\n" ); document.write( "The possible rational zeros (if any) are \r\n" ); document.write( "positive or negative fractions whose numerators\r\n" ); document.write( "are factors of 6 and whose denominators are\r\n" ); document.write( "factors of 5\r\n" ); document.write( "\r\n" ); document.write( "The factors of 6 are 1,2,3,6 (possible numerators)\r\n" ); document.write( "\r\n" ); document.write( "The factors of 5 are 1,5 (possible denominators)\r\n" ); document.write( "\r\n" ); document.write( "Possible fractions are 1/1, 1/5. 2/1, 2/5, 3/1, 3/5, 6/1, 6/5\r\n" ); document.write( "\r\n" ); document.write( "They reduce to 1, 1/5, 2, 2/5, 3, 3/5, 6, 6/5\r\n" ); document.write( "\r\n" ); document.write( "They could be positive or negative, so all possible rational\r\n" ); document.write( "zeros are:\r\n" ); document.write( "\r\n" ); document.write( "±1, ±1/5, ±2, ±2/5, ±3, ±3/5, ±6, ±6/5\r\n" ); document.write( "\r\n" ); document.write( "We will try the easiest one first, which is 1, using synthetic division\r\n" ); document.write( "to see if we get 0 remainder:\r\n" ); document.write( "\r\n" ); document.write( "1|5 -7 17 -21 6\r\n" ); document.write( " | 5 -2 15 -6 \r\n" ); document.write( " 5 -2 15 -6 0\r\n" ); document.write( "\r\n" ); document.write( "We're in luck. It worked. So we have now\r\n" ); document.write( "\r\n" ); document.write( "factored the original\r\n" ); document.write( "\r\n" ); document.write( "f(x)=5x4-7x³+17x²-21x+6\r\n" ); document.write( "\r\n" ); document.write( "as\r\n" ); document.write( "\r\n" ); document.write( "f(x)= (x - 1)(5x³ - 2x² + 15x - 6)\r\n" ); document.write( "\r\n" ); document.write( "Now we try to factor the polynomial in the second parentheses:\r\n" ); document.write( "\r\n" ); document.write( "First we change the parentheses ( ) to brackets [ ] so we can put\r\n" ); document.write( "parentheses inside with less confusion:\r\n" ); document.write( "\r\n" ); document.write( "f(x)= (x - 1)[5x³ - 2x² + 15x - 6]\r\n" ); document.write( "\r\n" ); document.write( "Out of the first two terms in the bracket we can factor out x²:\r\n" ); document.write( "\r\n" ); document.write( "f(x)= (x - 1)[x²(5x - 2) + 15x - 6]\r\n" ); document.write( "\r\n" ); document.write( "Out of the last two terms we can factor out 3:\r\n" ); document.write( "\r\n" ); document.write( "f(x)= (x - 1)[x²(5x - 2) + 3(5x - 2)]\r\n" ); document.write( "\r\n" ); document.write( "Now we have a common factor of (5x - 2)\r\n" ); document.write( "\r\n" ); document.write( "f(x) = (x - 1)[(5x - 2)(x² + 3)]\r\n" ); document.write( "\r\n" ); document.write( "We don't need the brackets anymore:\r\n" ); document.write( "\r\n" ); document.write( "f(x) = (x - 1)(5x - 2)(x² + 3)\r\n" ); document.write( "\r\n" ); document.write( "To find the zeros we set each factor = 0\r\n" ); document.write( "\r\n" ); document.write( "x - 1 = 0; 5x - 2 = 0; x² + 3 = 0\r\n" ); document.write( " x = 1; 5x = 2; x² = -3 _\r\n" ); document.write( " x = 2/5 x = ±i√3 (not real)\r\n" ); document.write( "\r\n" ); document.write( "So the correct choice is D\r\n" ); document.write( "\r\n" ); document.write( "Edwin\r \n" ); document.write( "\n" ); document.write( " \n" ); document.write( " |