document.write( "Question 788768: You are a life saver! I'm in Algebra three and we're going over polynomial functions. I'm supposed to find the end behavior, the y-intercept, the domain, the zeros, the multiplicity, and graph it. I understand some of it but I am mostly confused about end behavior and finding zeros when there are no 'real intercepts.'
\n" ); document.write( "The problem is f(x) = x^2 (x^2 + 1) (x + 4)
\n" ); document.write( "Could you help me with this?
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Algebra.Com's Answer #478616 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
You are a life saver! I'm in Algebra three and we're going over polynomial functions. I'm supposed to find the end behavior, the y-intercept, the domain, the zeros, the multiplicity, and graph it. I understand some of it but I am mostly confused about end behavior and finding zeros when there are no 'real intercepts.'
\n" ); document.write( "The problem is f(x) = x^2 (x^2 + 1) (x + 4)
\n" ); document.write( "Could you help me with this?\r
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document.write( "f(x) = x²(x² + 1)(x + 4)\r\n" );
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document.write( "To find the y-intercept, substitute x=0\r\n" );
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document.write( "f(0) = 0²(0² + 1)(0 + 4)\r\n" );
document.write( "f(0) = 0\r\n" );
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document.write( "So the y-intercept is (0,0)\r\n" );
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document.write( "The domain of every polynomial function is (-∞,∞), \"all real numbers\".\r\n" );
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document.write( "To find the zeros, set f(x) = 0 and solve for x\r\n" );
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document.write( "x²(x² + 1)(x + 4) = 0\r\n" );
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document.write( "Use the zero factor principle, that is, set each factor =0\r\n" );
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document.write( "x² = 0;   x² + 1 = 0;   x + 4 = 0\r\n" );
document.write( " x = 0;       x² = -1;      x = -4\r\n" );
document.write( "               x = ±V-1;\r\n" );
document.write( "               x = ±i;  \r\n" );
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document.write( "So there are four zeros, 2 real ones and two imaginary ones:\r\n" );
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document.write( "They are:\r\n" );
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document.write( "0, -4, i, -i\r\n" );
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document.write( "A zero R will have multiplicity of M if M is the largest positive \r\n" );
document.write( "integer such that (x-R)M is a factor of the polynomial.  \r\n" );
document.write( "This polynomial has the factor x² which is equivalent to (x-0)²,\r\n" );
document.write( "so the multiplicity of the zero 0 is 2, an even number, so the curve\r\n" );
document.write( "\"bounces off\" the x-axis at 0.  The factor (x+4)1 has \r\n" );
document.write( "exponent 1, an odd number so the zero -4 has multiplicity 2, an even\r\n" );
document.write( "number and the graph will cut through the x-axis at -4.\r\n" );
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document.write( "End behavior:\r\n" );
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document.write( "Rule: \r\n" );
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document.write( "multiply the right side all the way out:\r\n" );
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document.write( "f(x) = x²(x² + 1)(x + 4)\r\n" );
document.write( "f(x) = x5 + 4x4 + x3 + 4x2\r\n" );
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document.write( "Look at the leading term, the one with the largest exponent, 1x5 \r\n" );
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document.write( "Rules:\r\n" );
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document.write( "1. If its coefficient is positive the curve goes UP on the extreme right.\r\n" );
document.write( "2. If its coefficient is negative the curve goes DOWN on the extreme right.\r\n" );
document.write( "3. If the largest exponent (the degree) is even, the extreme left hand behavior\r\n" );
document.write( "   is the SAME as the right hand behavior.\r\n" );
document.write( "4. If the exponent (the degree) is odd, the extreme left hand behavior is the\r\n" );
document.write( "   OPPOSITE of the right hand behavior.\r\n" );
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document.write( "Since  1x5 has a positive coefficient, the extreme right hand\r\n" );
document.write( "behavior is UP on the right.\r\n" );
document.write( "Since  1x5 has an ODD exponent, the extreme left hand behavior is\r\n" );
document.write( "the OPPOSITE, so the extreme left-hand behavior is DOWN of the extreme left.   \r\n" );
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document.write( "Here is the graph:\r\n" );
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document.write( "\"graph%28400%2C400%2C-5%2C3%2C-100%2C100%2Cx%5E2%28x%5E2%2B1%29%28x%2B4%29%29\"\r\n" );
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document.write( "Notice the on the far right the curve goes UP, and on the far left it goes\r\n" );
document.write( "DOWN.  Notice that the graph \"bounces off\" the x axis at the zero 0, and cuts\r\n" );
document.write( "through the x-axis at the zero -4.\r\n" );
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document.write( "Edwin
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