document.write( "Question 637123: Please help me solve this problem: (x^3 - 13x^2 - 41x + 4) / (x - 7)\r
\n" ); document.write( "\n" ); document.write( "Right now I have the answer as x^2 -6x + 48\r
\n" ); document.write( "\n" ); document.write( "I don't think what I have so far is right, and I don't know how to finish it. Please help, thank you. \n" ); document.write( "
Algebra.Com's Answer #401531 by DrBeeee(684)\"\" \"About 
You can put this solution on YOUR website!
Your suspection that your quadratic is wrong couldn't be truer. The product of the two constant terms, -7 and 48 is not equal to 4 as it should be. As to how you proceed to factor the trinomial, you can take solace in the fact that a trinomial (or any odd powered polynomial) has at least one real root. This is because complex roots occur in pairs or an even number.
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\n" ); document.write( "Since we have at least one real root, the function must cross the x-axis at least once. I use the trial/average technique to find the real value of x that makes the trinomial equal to zero. That is, we find out where the given trinomial crosses the x-axis. Let's do it, OK?\r
\n" ); document.write( "\n" ); document.write( "Let f(x) = x^3 - 13*x^2 - 41*x + 4.
\n" ); document.write( "I start with x = 0
\n" ); document.write( "Then f(0) = 0 - 13*0 - 41*0 + 4 = 4 > 0
\n" ); document.write( "Now try x = 1
\n" ); document.write( "f(1) = 1 - 13 - 41 + 4 = -49 < 0
\n" ); document.write( "Now we know that f(x) crossed the x-axis somewhere between 0 and 1 (because f(0) is + and f(1) is negative). Now calculate the average of 0 and 1. This is .5, so try x = .5
\n" ); document.write( "f(.5) = 0.125 - 3.25 - 20.5 + 4 = - 19.625 < 0.
\n" ); document.write( "Now 0 < root < .5, so try the average or 0.25 etc.
\n" ); document.write( "f(0.25) < 0
\n" ); document.write( "f(0.125) < 0
\n" ); document.write( "f(0.0625) > 0
\n" ); document.write( "Root lies between 1/16 < x < 1/8
\n" ); document.write( "f(3/32) = +0.043 > 0
\n" ); document.write( "f(7/64) = -0.638 < 0
\n" ); document.write( "After many tries, I determined that
\n" ); document.write( "f(0.09473600) = 0.00000042 which is very close to zero.
\n" ); document.write( "The first order factor is
\n" ); document.write( "(x - 0.094736)
\n" ); document.write( "FOIL this with (x^2 + b*x + c) and set the product equal to the given trinomial. Then solve for b and c by equating the coefficients of x^2, x^1 and x^0. This evaluation will give
\n" ); document.write( " b = -12.905264 and
\n" ); document.write( " c = -42.2226
\n" ); document.write( "The factorization is
\n" ); document.write( "(x - 0.094736)(x^2 -12.905264x -42.2226) = x^3 -13x^2 - 41x +4
\n" ); document.write( "The quadratic can be factored by using the quadratic equation. To help you out, here's my version
\n" ); document.write( "x = (-b/2 +/-sqrt((b/2)^2-a*c))/a
\n" ); document.write( "where a = 1, b = -12.905264 and c = -42.2226. Plug and grind gives
\n" ); document.write( "x = 6.452632 +/- 9.15746 or
\n" ); document.write( "x = -2.704828, 15.610092 with factors
\n" ); document.write( "(x + 2.704828)(x - 15.610092) which when FOILed yields
\n" ); document.write( "x^2 - 12.90526x -4202226 as given above.
\n" ); document.write( "The final factorization is
\n" ); document.write( "(x - 0.094736)(x + 2.704828)(x - 15.610092) which equals
\n" ); document.write( "x^3 -13x^2 -41x +4
\n" ); document.write( "Amen
\n" ); document.write( "PS some numbers may not agree exactly because of all the decimals values needed. \n" ); document.write( "
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