document.write( "Question 1090083: The polynomial $f(x)$ has degree 3. If $f(-1) = 15$, $f(0)= 0$, $f(1) = -5$, and $f(2) = 12$, then what are the $x$-intercepts of the graph of $f$? Please explain your work thoroughly so I can understand. Thanks!
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Algebra.Com's Answer #704506 by ikleyn(52812) $\"\"$ $\"About$

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document.write( "Let  f(x) = ax^3 + bx^2 + cx + d  with unknown coefficients  a, b, c and d.\r\n" );
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document.write( "Since f(0) = 0, it implies d = 0   (to see it, simply substitute x= 0 into the polynomial).\r\n" );
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document.write( "So, you need to determine a, b and c.\r\n" );
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document.write( "f(-1) = 15  ====> -a +  b -  c = 15,     (1)\r\n" );
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document.write( "f(1)  = -5  ====>  a +  b +  c = -5,     (2)\r\n" );
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document.write( "f(2)  = 12  ====> 8a + 4b + 2c = 12.     (3)\r\n" );
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document.write( "Add equations (1) and (2). You will get   2b = 15 + (-5) = 10  ====>  b = 5.\r\n" );
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document.write( "Then equations (2) and (3) take the form\r\n" );
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document.write( " a +   5 +  c = -5     (4)    (instead of (2))\r\n" );
document.write( "8a + 4*5 + 2c = 12     (5)    (instead of (3))\r\n" );
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document.write( "or, which is the same\r\n" );
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document.write( " a + c = -10,          (4')\r\n" );
document.write( "4a + c =  -4,          (5')\r\n" );
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document.write( "Subtract (4') from (5') to get 3a = 6,   a = 2.\r\n" );
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document.write( "Then  from (4')  c = -12.\r\n" );
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document.write( "Thus the polynomial is  f(x) =  =  =  (factor ! ) =  = .\r\n" );
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document.write( "Thus the x-intercepts are , 0 (given !)  and  -4.\r\n" );
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