document.write( "Question 381141: The question wants you to find the complex zeros and factor the polynomial in standard form: F(X) = x^3 - 1. \n" ); document.write( "

Algebra.Com's Answer #270408 by neatmath(302) $\"\"$ $\"About$

You can put this solution on YOUR website!

We can easily see that $\"f%28x%29=x%5E3-1\"$ is the difference of two cubes!
\r
\n" ); document.write( "\n" ); document.write( " $\"f%28x%29=x%5E3-1\"$
\r
\n" ); document.write( "\n" ); document.write( " $\"f%28x%29=%28x-1%29%28x%5E2%2Bx%2B1%29\"$
\r
\n" ); document.write( "\n" ); document.write( " $\"0=%28x-1%29%28x%5E2%2Bx%2B1%29\"$ set f(x) equal to zero, so we have
\r
\n" ); document.write( "\n" ); document.write( " $\"%28x-1%29=0\"$ or $\"%28x%5E2%2Bx%2B1%29=0\"$
\r
\n" ); document.write( "\n" ); document.write( " $\"x-1=0\"$
\r
\n" ); document.write( "\n" ); document.write( " $\"x=1\"$ or
\r
\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2Bx%2B1%29=0\"$
\r
\n" ); document.write( "\n" ); document.write( "This will involve imaginary numbers because there are NO real numbers which satisfy this equation, but we can use the quadratic formula to find the complex solutions involving imaginary numbers:
\r
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "\n" ); document.write( "

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation $\"ax%5E2%2Bbx%2Bc=0\"$ (in our case $\"1x%5E2%2B1x%2B1+=+0\"$ ) has the following solutons:
\n" ); document.write( "
\n" ); document.write( " $\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"$
\n" ); document.write( "
\n" ); document.write( " For these solutions to exist, the discriminant $\"b%5E2-4ac\"$ should not be a negative number.
\n" ); document.write( "
\n" ); document.write( " First, we need to compute the discriminant $\"b%5E2-4ac\"$ : $\"b%5E2-4ac=%281%29%5E2-4%2A1%2A1=-3\"$ .
\n" ); document.write( "
\n" ); document.write( " The discriminant -3 is less than zero. That means that there are no solutions among real numbers.

\n" ); document.write( " If you are a student of advanced school algebra and are aware about imaginary numbers, read on.

\n" ); document.write( "
\n" ); document.write( " In the field of imaginary numbers, the square root of -3 is + or - $\"sqrt%28+3%29+=+1.73205080756888\"$ .
\n" ); document.write( "
\n" ); document.write( " The solution is

\n" ); document.write( "
\n" ); document.write( " Here's your graph:
\n" ); document.write( " $\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B1%2Ax%2B1+%29\"$

\n" ); document.write( "
\n" ); document.write( "

\n" );