document.write( "Question 74704: I am having problems solving this equation \"x%5E3-6x%5E2-15x-8=0\"
\n" ); document.write( "This is what I have done:
\n" ); document.write( "x^3-6x^2-15x-8=0
\n" ); document.write( "x^3-6x^2-15x=8
\n" ); document.write( "x(x^2-6x-15)=8
\n" ); document.write( "i tried to factor x^2-6x-15 by completing square like this, since it is not a perfect squared trinomial.
\n" ); document.write( "x(x^2-6x+9)-15-9=8
\n" ); document.write( "x(x-3)^2-32=0
\n" ); document.write( "I am stucked here. How can I find the roots of the above polynomial?
\n" ); document.write( "Someone showed me the factored answer
\n" ); document.write( "(x+1)(x+1)(x-8)=0 and then the roots are x=-1, 8
\n" ); document.write( "If you multiply the above factors it will give you the above polynomial. And the roots satisfy the equation. But how can i get this answer?
\n" ); document.write( "I appreciate your help. Thank you very much.
\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #53689 by Earlsdon(6294)\"\" \"About 
You can put this solution on YOUR website!
Solve:
\n" ); document.write( "\"x%5E3-6x%5E2-15x-8+=+0\"
\n" ); document.write( "You could try using the \"Factor theorem\" (related to the \"remainder theorem\")which states that...\"if (x-a) is a factor of the polynomial, there will be no remainder (when the polynomial is divided by (x-a)), therefore f(a) = 0.
\n" ); document.write( "How do you apply this to factoring your polynomial?
\n" ); document.write( "You will need to choose values that make sense as possible factors.
\n" ); document.write( "Using the constant term in the polynomial, (-8), what are the factors of -8?
\n" ); document.write( "1(-8) = -8
\n" ); document.write( "(-1)(8) = -8
\n" ); document.write( "2(-4) = -8
\n" ); document.write( "(-2)(4) = -8
\n" ); document.write( "So, the sensible choices are:
\n" ); document.write( "1, -1, 2, -2, 4, -4, 8, -8
\n" ); document.write( "Start with 1.
\n" ); document.write( "f(1) = 1^3-6(1)^2-15(1)-8
\n" ); document.write( "f(1) = 1-6-15-8
\n" ); document.write( "f(1) = -28 Since this is not = 0, then (x-1) is not a factor.
\n" ); document.write( "Try -1.
\n" ); document.write( "f(-1) = (-1)^3-6(-1)^2-15(-1)-8
\n" ); document.write( "f(-1) = -1-6+15-8
\n" ); document.write( "f(-1) = -15+15
\n" ); document.write( "f(-1) = 0, so (x+1) is a factor.
\n" ); document.write( "Now try 8 because you know that (-1)(8) = -8, the constant term.
\n" ); document.write( "It also makes sense that (x+1) is a repeated factor because of the fact that (-1)(8) = -8
\n" ); document.write( "f(8) = 8^3-6(8)^2-15(8)-8
\n" ); document.write( "f(8) = 512-384-120-8
\n" ); document.write( "f(8) = 512-512
\n" ); document.write( "f(8) = 0 so (x-8) is a factor.\r
\n" ); document.write( "\n" ); document.write( "So, now we have:
\n" ); document.write( "(x+1)(x+1)(x-8) = 0
\n" ); document.write( "x+1 = 0, so x = -1
\n" ); document.write( "x+1 = 0, so x = -1
\n" ); document.write( "x-8 = 0, so x = 8
\n" ); document.write( "Check:
\n" ); document.write( "\"%28x%2B1%29%28x%2B1%29%28x-8%29+=+%28x%2B1%29%28x%5E2-7x-8%29\" = \"x%5E3-7x%5E2-8x%2Bx%5E2-7x-8+=+x%5E3-6x%5E2-15x-8\"
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