document.write( "Question 12742: The directions say to factor as completely as possible
\n" ); document.write( "(x-5)^3 -1\r
\n" ); document.write( "\n" ); document.write( "The answer is (x-6) (x^2 -9x +21) \r
\n" ); document.write( "\n" ); document.write( "I cannot figure out how to get this answer. Can you please show the steps? \n" ); document.write( "
Algebra.Com's Answer #6461 by glabow(165)\"\" \"About 
You can put this solution on YOUR website!
First, expand the expression \"%28x-5%29%5E3\" and subtract 1.
\n" ); document.write( "This is \"%28x-5%29%28x-5%29%28x-5%29+-1\"
\n" ); document.write( "This gives \"%28x%5E2-10x%2B25%29%28x-5%29+-1\"
\n" ); document.write( "Then \"%28x%5E3%29-%2810x%5E2%29%2B%2825x%29-%285x%5E2%29%2B%2850x%29-125+-1\"
\n" ); document.write( "And finally \"x%5E3-15x%5E2%2B75x-126\"
\n" ); document.write( "Now you factor this expression. This is done most easily by using the fact that IF IT FACTORS, the factors must be factors of 126/1 [where the numerator is the coefficient of the last term and the denominator is the coefficient of the highest term]. This only applies when the coefficients of the terms are integers. So the possible factors of this expression are 126, 63, 31, 6, 2, and 1.
\n" ); document.write( "You check them by putting them into the expression as the value of x and see if the expression evaluates to 0. If a number n results in the expression evaluating to 0, then (x-a) is a factor. (Do you see why? x-a will equal 0, and 0 times the other factors will always be 0.)
\n" ); document.write( "The simple way to check these values in the expression is by synthetic substitution, which you should either look up or review if it was taught.
\n" ); document.write( "Anyway, checking with 6 results in \"remainder\" coefficients of 1, -9, 21, and 0. This means 6 is a factor, and that (x-6) evenly divides the expression. The remainder after dividing the expression by (x-6) will be \"x%5E2+-9x%2B21\" and that leaves the factors as \"%28x-6%29%28x%5E2-9x%2B21%29\"
\n" ); document.write( "You then tackle finding factors for \"x%5E2-9x%2B21\"
\n" ); document.write( "The only possible factors are 21, 7, 3, and 1. And none of them divide the expression evenly. So, there are no integer factors of the expression.
\n" ); document.write( "The simplest factoring of the original expression is \"%28x-6%29%28x%5E2-9x%2B21%29\"
\n" ); document.write( "[A side note: this is a fairly complex problem unless you know the basic factor theorem and synthetic substitution. If those have not been taught to you it becomes a very difficult problem to solve.]
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