document.write( "Question 68111: Find the polynomial f(x) of degree three that has zeroes at 1, 2 and 4 such that f(0)=-16.\r
\n" ); document.write( "\n" ); document.write( "The degree three polynomial f(x) with real coefficients and leading coefficient 1, has 4 and 3 + i among its roots. Express f(x) as a product of linear and quadratic polynomials with real coefficients.\r
\n" ); document.write( "\n" ); document.write( "Given that (3x-a)(x-2)(x-7)=3x^2-32x^2+81x-70, determine the value of a.\r
\n" ); document.write( "\n" ); document.write( "Find all roots of the polynomial x^3-x^2+16x-16.\r
\n" ); document.write( "\n" ); document.write( "Find the vertical asymptote of the rational function f(x)=3x-12/4x-2.\r
\n" ); document.write( "\n" ); document.write( "Find the horizontal asymptote of the rational function f(x)=8x-12/4x-2.\r
\n" ); document.write( "\n" ); document.write( "Studying for final exam please respond ASAP. Thanks for your assistance. \n" ); document.write( "
Algebra.Com's Answer #48403 by Edwin McCravy(20059)\"\" \"About 
You can put this solution on YOUR website!
Find the polynomial f(x) of degree three that has zeroes \r\n" );
document.write( "at 1, 2 and 4 such that f(0)=-16.\r\n" );
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document.write( "Two ways to do it:\r\n" );
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document.write( "First way:\r\n" );
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document.write( "Rule 1: If a polynomial f(x) has roots r1, r2, r3, ···, rn\r\n" );
document.write( "and leading coefficient a, then\r\n" );
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document.write( "f(x) = a(x-r1)(x-r2)···(x-rn)\r\n" );
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document.write( "So for this problem:\r\n" );
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document.write( "f(x) = a(x - 1)(x - 2)(x - 4)\r\n" );
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document.write( " -16 = a(0 - 1)(0 - 2)(0 - 4)\r\n" );
document.write( " -16 = a(-1)(-2)(-4)\r\n" );
document.write( " -16 = -8a\r\n" );
document.write( "   2 = a\r\n" );
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document.write( "f(x) = 2(x - 1)(x - 2)(x - 4)\r\n" );
document.write( "f(x) = 2(x - 1)(x² - 6x + 8)\r\n" );
document.write( "f(x) = 2(x³ - 6x² + 8x - x² + 6x - 8)\r\n" );
document.write( "f(x) = 2(x³ - 7x² + 14x - 8)\r\n" );
document.write( "f(x) = 2x³ - 14x² + 28x - 16\r\n" );
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document.write( "Here's the second method: \r\n" );
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document.write( "Let the polynomial be \r\n" );
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document.write( "f(x) = ax³ + bx² + cx + d\r\n" );
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document.write( "Then plugging in the values for x, and\r\n" );
document.write( "setting it to -16 when 0 is plugged in,\r\n" );
document.write( "and 0 when the others are plugged in:\r\n" );
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document.write( "f(0) = a(0)³ + b(0)² + c(0) + d = -16\r\n" );
document.write( "f(1) = a(1)³ + b(1)² + c(1) + d = 0\r\n" );
document.write( "f(2) = a(2)³ + b(2)² + c(2) + d = 0\r\n" );
document.write( "f(4) = a(4)³ + b(4)² + c(4) + d = 0\r\n" );
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document.write( "The above simplifies to this system\r\n" );
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document.write( "                              d = -16\r\n" );
document.write( "                  a + b + c + d = 0\r\n" );
document.write( "               8a + 4b + 2c + d = 0 \r\n" );
document.write( "             64a + 16b + 4c + d = 0\r\n" );
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document.write( "Substitute =16 for d in the bottom three:\r\n" );
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document.write( "                  a + b + c - 16 = 0\r\n" );
document.write( "               8a + 4b + 2c - 16 = 0 \r\n" );
document.write( "             64a + 16b + 4c - 16 = 0\r\n" );
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document.write( "                  a + b + c = 16 \r\n" );
document.write( "               8a + 4b + 2c = 16  \r\n" );
document.write( "             64a + 16b + 4c = 16\r\n" );
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document.write( "Solve that system of equations as you get\r\n" );
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document.write( "a = 2, b = -14, c = 28\r\n" );
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document.write( "So the polynomial\r\n" );
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document.write( "f(x) = ax³ + bx² + cx + d becomes\r\n" );
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document.write( "f(x) = 2x³ - 14x + 28x - 16               \r\n" );
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document.write( "The degree three polynomial f(x) with real coefficients and \r\n" );
document.write( "leading coefficient 1, has 4 and 3 + i among its roots. \r\n" );
document.write( "Express f(x) as a product of linear and quadratic polynomials \r\n" );
document.write( "with real coefficients.\r\n" );
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document.write( "Rule 2:\r\n" );
document.write( "If a polynomial has real coefficients, then if a complex\r\n" );
document.write( "number a + bi is a root, then so is its conjugate a - bi\r\n" );
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document.write( "So the roots are 4, 3+i, 3-i\r\n" );
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document.write( "Using rule 1\r\n" );
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document.write( "f(x) = 1(x - 4)[x - (3+i)][x - (3-i)]\r\n" );
document.write( "f(x) = (x - 4)[x² - (3-i)x - (3+i)x + (3+i)(3-i)]\r\n" );
document.write( "f(x) = (x - 4)[x² - 3x + ix - 3x - ix + 9 - i²]\r\n" );
document.write( "f(x) = (x - 4)[x² - 6x + 9 - i²]\r\n" );
document.write( "Since i² = -1, substitute -1 for i²\r\n" );
document.write( "f(x) = (x - 4)[x² - 6x + 9 - (-1)]\r\n" );
document.write( "f(x) = (x - 4)(x² - 6x + 9 + 1]\r\n" );
document.write( "f(x) = (x - 4)(x² - 6x + 10)\r\n" );
document.write( "f(x) = x³ - 6x² + 10x - 4x² + 24x - 40 \r\n" );
document.write( "f(x) = x³ - 10x² + 34x - 40\r\n" );
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document.write( "Given that (3x-a)(x-2)(x-7)=3x³-32x²+81x-70, determine the \r\n" );
document.write( "value of a.\r\n" );
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document.write( "Since this is an identity, we may substitute any number for x\r\n" );
document.write( "and it will be true.  It wouldn't do any good to substitute a \r\n" );
document.write( "root 2 or 7 since that would just give 0 = 0. So let's \r\n" );
document.write( "substitute x = 0, since that is the easiest number to \r\n" );
document.write( "substitute, and 0 is not a root:\r\n" );
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document.write( "          (3x-a)(x-2)(x-7) = 3x³-32x²+81x-70\r\n" );
document.write( "        [3(0)-a](0-2)(0-7) = 3(0)³-32(0)²+81(0)-70\r\n" );
document.write( "              (-a)(-2)(-7) = -70\r\n" );
document.write( "                      -14a = -70 \r\n" );
document.write( "                         a = 5       \r\n" );
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document.write( "Find all roots of the polynomial x^3-x^2+16x-16.\r\n" );
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document.write( "Set it = 0\r\n" );
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document.write( "   x³ - x² + 16x - 16 = 0\r\n" );
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document.write( "Factor by grouping:\r\n" );
document.write( "Factor x² out of the first two terms\r\n" );
document.write( "and 16 out of the last two terms:\r\n" );
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document.write( "x²(x - 1) + 16(x - 1) = 0\r\n" );
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document.write( "Factor out common factor (x - 1)\r\n" );
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document.write( "(x - 1)(x² + 16) = 0\r\n" );
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document.write( "Set each factor = 0\r\n" );
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document.write( "x - 1 = 0 \r\n" );
document.write( "    x = 1\r\n" );
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document.write( "x² + 16 = 0\r\n" );
document.write( "     x² = -16  \r\n" );
document.write( "      x = ±Ö-16\r\n" );
document.write( "      x = ±4i\r\n" );
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document.write( "Find the vertical asymptote of the rational \r\n" );
document.write( "function f(x)=(3x-12)/(4x-2).\r\n" );
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document.write( "Set denominator = 0\r\n" );
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document.write( "4x - 2 = 0\r\n" );
document.write( "    4x = 2\r\n" );
document.write( "     x = 1/2\r\n" );
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document.write( "Find the horizontal asymptote of the rational function \r\n" );
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document.write( "f(x) = \"%288x-12%29%2F%284x-2%29\".\r\n" );
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document.write( "Rule: Divide every term on top and bottom by the greatest\r\n" );
document.write( "power of x\r\n" );
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document.write( "f(x) = \"%28%288x%29%2Fx+-+12%2Fx%29%2F%28%284x%29%2Fx+-+2%2Fx%29\" \r\n" );
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document.write( "Simplify\r\n" );
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document.write( "f(x) = \"%288+-+12%2Fx%29%2F%282+-+2%2Fx%29\"\r\n" );
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document.write( "As x grows very large in absolute value, the fractions \r\n" );
document.write( "get extremely small, and become negligible, so f(x)\r\n" );
document.write( "approaches the fraction 8/2 or 4, so the horizontal\r\n" );
document.write( "asymptote is y = 4.\r\n" );
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document.write( "Or you can learn the rule that if the numerator and\r\n" );
document.write( "denominator have the same degree, the horizontal \r\n" );
document.write( "asymptote's equation is the quotient of the leading\r\n" );
document.write( "coefficients, y = 8/4  or y = 2 \r\n" );
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document.write( "Edwin
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