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Algebra.Com's Answer #840682 by math_tutor2020(3817)![\"\"](\"/images/stars/stars-13.gif\")   You can put this solution on YOUR website! \n" ); document.write( "Let \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "The original equation that your teacher gave you turns into \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Multiply both sides by the LCD w^2 to clear out the fractions. \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "Use the rational root theorem to check the factors of the last term (1) \n" ); document.write( "The possible rational roots would be: 1 or -1 \n" ); document.write( "Plug in w = -1 to get a nonzero result \n" ); document.write( "But w = 1 would lead to zero, which shows w = 1 is a root.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "The root w = 1 means w-1 is a factor of w^4 + w^3 - 4w^2 + w + 1\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Use polynomial long division, or synthetic division, to find that \n" ); document.write( " \n" ); document.write( "or simply \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Then notice how w = 1 is also a root of w^3+2w^2-2w-1 due to the rational root theorem.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "w = 1 is a repeated root of w^4 + w^3 - 4w^2 + w + 1 \n" ); document.write( "So (w-1)^2 is a factor of w^4 + w^3 - 4w^2 + w + 1\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Turns out that \n" ); document.write( " \n" ); document.write( "factors to \n" ); document.write( " \n" ); document.write( "Polynomial long division can be used to find this.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "From here, use the quadratic formula to solve w^2+3w+1 = 0 \n" ); document.write( "I'll skip steps. \n" ); document.write( "The results are: \n" ); document.write( "They approximate to w = -2.618 and w = -0.382 respectively.\r \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "In summary so far: \n" ); document.write( "The four roots of w^4 + w^3 - 4w^2 + w + 1 are \n" ); document.write( "w = 1 \n" ); document.write( "w = 1 \n" ); document.write( "w = -2.618 (approximate) \n" ); document.write( "w = -0.382 (approximate)\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Recall we made w = sin(x) \n" ); document.write( "w = 1 leads to sin(x) = 1 and this partial solution set: \n" ); document.write( "x = pi/2 + 2pi*n \n" ); document.write( "where n is an integer. \n" ); document.write( "I'm assuming you are in radian mode. \n" ); document.write( "If you are in degree mode, then replace pi/2 with 90 and replace 2pi with 360.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "w = -2.618 does not lead to any real number solutions for x because the smallest sin(x) can get is -1. \r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "w = -0.382 does lead to solutions for x since -0.382 is between -1 and 1. \n" ); document.write( "I'll let you solve sin(x) = -0.382 \n" ); document.write( "There are two sub-branches here. \n" ); document.write( " \n" ); document.write( " |