document.write( "Question 520002: 1)consider the polynomial
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Algebra.Com's Answer #345919 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
1)consider the polynomial
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document.write( "Assume that the other factor of p(x) is x-r.  Then\r\n" );
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document.write( "p(x) = x³ + ax² + bx - 12 = (x + 3)(x - 4)(x - r) = (x² - x - 12)(x - r) =\r\n" );
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document.write( "x³ - rx - x² + rx - 12x + 12r\r\n" );
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document.write( "x³ + ax² + bx - 12 = x³ - rx² - x² + rx - 12x + 12r\r\n" );
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document.write( "x³ + ax² + bx - 12 = x³ - (r + 1)x² + (r - 12)x + 12r\r\n" );
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document.write( "Equate coefficients of like powers of x:\r\n" );
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document.write( "a = -(r + 1)\r\n" );
document.write( "b = (r - 12)\r\n" );
document.write( "-12 = 12r which tells us that r = -1\r\n" );
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document.write( "a = -(r + 1) = -(-1+1) = 0\r\n" );
document.write( "b = (r - 12) = -1 - 12 = -13\r\n" );
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document.write( "p(x) = x³ + ax² + bx - 12 = (x + 3)(x - 4)(x - r) = \r\n" );
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document.write( "(x + 3)(x - 4)[x -(-1)] =\r\n" );
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document.write( "(x + 3)(x - 4)(x + 1)\r\n" );
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\n" ); document.write( "2)the polynomial p(x)=2x^3-ax^2+bx+48 has (x+4) as a repeated factor , find the values of a and b.\r
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document.write( "Assume that the factors are (x - 4), (x - 4), (x - r), and 2.\r\n" );
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document.write( "Then\r\n" );
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document.write( "p(x) = 2x³ + ax² + bx - 12 = (x - 4)(x - 4)2(x - r)\r\n" );
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document.write( "Notice that I put in the factor 2 because the first coefficient is 2\r\n" );
document.write( "and that 2 is necessary to make the terms in x³ the same.\r\n" );
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document.write( "p(x) = (x - 4)(x - 4)2(x - r)\r\n" );
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document.write( "Multiply that all the way out and get\r\n" );
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document.write( "p(x) = 2x³ + ax² + bx - 12 \r\n" );
document.write( "= 2x³ - 2rx² - 16x² + 16rx + 32x - 32r =\r\n" );
document.write( "2x³ + (-2r-16)x² + (16r+32)x - 32r \r\n" );
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document.write( "Now equate the coeficcients of\r\n" );
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document.write( "2x³ + ax² + bx - 12\r\n" );
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document.write( "and\r\n" );
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document.write( "2x³ + (-2r-16)x² + (16r+32)x - 32r\r\n" );
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document.write( "Equating the coefficients of x²:\r\n" );
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document.write( "a = -2r - 16\r\n" );
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document.write( "Equating the coefficients of x\r\n" );
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document.write( "b = 16r + 32\r\n" );
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document.write( "Equating the constant term:\r\n" );
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document.write( "-32r = -12\r\n" );
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document.write( "Solving for r:\r\n" );
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document.write( "r = \"%28-12%29%2F%28-32%29\" = {{3/8}}}\r\n" );
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document.write( "Substituting r = \"3%2F8\" in\r\n" );
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document.write( "b = 16r+32\r\n" );
document.write( "b = 16(\"3%2F8\")+32\r\n" );
document.write( "b = 38\r\n" );
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document.write( "Substituting r = \"3%2F8\" in\r\n" );
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document.write( "a = -2r - 16\r\n" );
document.write( "a = -2\"3%2F8\" - 16\r\n" );
document.write( "a = \"-3%2F4\" - 16\r\n" );
document.write( "a = \"-67%2F4\"\r\n" );
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document.write( "So all you want is a = \"-67%2F4\" and b = 38.\r\n" );
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document.write( "Checking: the polynomial function you want is \r\n" );
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document.write( "p(x) = 2x³ + \"-67%2F4\"x² + 38x - 12\r\n" );
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document.write( "and it \"factorises\" [We say \"factors\" in the US, we don't say \"factorises\",\r\n" );
document.write( "like we say \"color\", \"flavor\" and \"center\", not \"colour\", \"flavour\", and \"centre\"] :)\r\n" );
document.write( "as:\r\n" );
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document.write( "p(x) = (x - 4)(x - 4)2(x - r)\r\n" );
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document.write( "p(x) = (x - 4)(x - 4)2(x - \"3%2F8\")\r\n" );
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document.write( "p(x) = (x - 4)²(2)(x - \"3%2F8\")\r\n" );
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document.write( "or you can multiply the 2 into the parentheses\r\n" );
document.write( "that follows it and get\r\n" );
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document.write( "p(x) = (x - 4)²(2x - \"3%2F4\")\r\n" );
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document.write( "Or maybe you want to get a least common\r\n" );
document.write( "denominator in the parentheses:\r\n" );
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document.write( "p(x) = (x - 4)²(\"8x%2F4\" - \"3%2F4\")\r\n" );
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document.write( "p(x) = (x - 4)²(\"%288x-3%29%2F4\")\r\n" );
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document.write( "and put the \"1%2F4\" out in front and get:\r\n" );
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document.write( "\"1%2F4\"(x - 4)²(8x - 3)\r\n" );
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document.write( "The graph of p(x) is given below:\r\n" );
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document.write( "Edwin
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