document.write( "Question 548737: I need to find all zeros\r
\n" ); document.write( "\n" ); document.write( "13x^3-185x^2+608x-580 \n" ); document.write( "

Algebra.Com's Answer #357529 by KMST(5328) $\"\"$ $\"About$

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I would first look for a rational zero and divide.
\n" ); document.write( "A rational zero would be a fraction whose numerator divides 580 and whose denominator divides 13.
\n" ); document.write( "The prime factorization of 580 is $\"580=2%5E2%2A5%2A29\"$
\n" ); document.write( "That means it has 12 positive divisors: 1,2,4,5,10,20,29,58,116,145,290, and 580.
\n" ); document.write( "Considering the possible denominators 1 and 13 (divisors of 13), that would make 24 positive rational numbers. With the corresponding 24 negative ones, that's 48 possible zeros to try.
\n" ); document.write( "Luckily, 2 is a zero of that polynomial. Dividing by $\"x-2\"$
\n" ); document.write( "I find $\"13x%5E3-185x%5E2%2B608x-580=%28x-2%29%2813x%5E2-159x%2B290%29\"$
\n" ); document.write( "Applying the quadratic formula to $\"13x%5E2-159x%2B290=0\"$
\n" ); document.write( "we can find any remaining zeros.
\n" ); document.write( "

\n" ); document.write( "That gives us $\"x=10\"$ and $\"x=29%2F13\"$ as the last two zeros. \n" ); document.write( "

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