document.write( "Question 825731: Find all zeros of the function f(x)=x^2(x-3)(x^3-8) \n" ); document.write( "

Algebra.Com's Answer #497569 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"f%28x%29=x%5E2%28x-3%29%28x%5E3-8%29\"$
\n" ); document.write( "Finding zeros of a polynomial usually involves factoring the polynomial. This function is already partially factored so some of our work has already been done.

\n" ); document.write( "The factor at the end, $\"x%5E3-8\"$ , is a difference of cubes, $\"x%5E3-2%5E3\"$ . So we can use the difference of cubes pattern, $\"a%5E3-b%5E3=%28a-b%29%28a%5E2%2Bab%2Bb%5E2%29\"$ , to factor it:
\n" ); document.write( " $\"f%28x%29=x%5E2%28x-3%29%28x-2%29%28x%5E2%2B2x%2B4%29\"$
\n" ); document.write( "The last factor will not factor further. But it is a quadratic so we can use the quadratic formula to find its roots:
\n" ); document.write( " $\"x+=+%28-%282%29+%2B-+sqrt%28%282%29%5E2-4%281%29%284%29%29%29%2F2%281%29\"$
\n" ); document.write( "Simplifying...
\n" ); document.write( " $\"x+=+%28-2+%2B-+sqrt%284-4%281%29%284%29%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%28-2+%2B-+sqrt%284-16%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%28-2+%2B-+sqrt%28-12%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%28-2+%2B-+sqrt%28-1%2A4%2A3%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%28-2+%2B-+sqrt%28-1%29%2Asqrt%284%29%2Asqrt%283%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%28-2+%2B-+i%2A2%2Asqrt%283%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%28-2+%2B-+2i%2Asqrt%283%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%282%28-1+%2B-+i%2Asqrt%283%29%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%28cross%282%29%28-1+%2B-+i%2Asqrt%283%29%29%29%2Fcross%282%29\"$
\n" ); document.write( " $\"x+=+-1+%2B-+i%2Asqrt%283%29\"$
\n" ); document.write( "which is short for:
\n" ); document.write( " $\"x+=+-1+%2B+i%2Asqrt%283%29\"$ or $\"x+=+-1+-+i%2Asqrt%283%29\"$
\n" ); document.write( "These are two of the roots of f(x).

\n" ); document.write( "We will get the remaining roots from the other factors:
\n" ); document.write( " $\"f%28x%29=x%5E2%28x-3%29%28x-2%29%28x%5E2%2B2x%2B4%29\"$
\n" ); document.write( "The other roots will be the values for x that make a factor zero. For the first factor, $\"x%5E2\"$ , we get a root of x = 0. And since x is a factor twice in $\"x%5E2\"$ , zero counts as a root twice! (This is called a double root.)

\n" ); document.write( "For the other factors we should get roots of x = 3 and x = 2.

\n" ); document.write( "Altogether, the roots of f(x) are: 0, 0, 2, 3, $\"-1+%2B+i%2Asqrt%283%29\"$ and $\"-1+-+i%2Asqrt%283%29\"$

\n" ); document.write( "P.S. This is a polynomial, not a rational function. Please post in an appropriate category. \n" ); document.write( "

\n" );