document.write( "Question 153021: Find the Inverse of $\"+f%28x%29=3%2Bx%5E3+\"$ \r
\n" ); document.write( "\n" ); document.write( "Use Descarte's Rule of Signes to determine the number of positive and negative real zeros of the polynomial $\"+p%28x%29+=+2x%5E4%2B3x%5E3-2x%5E2%2Bx-2+\"$ can have\r
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Algebra.Com's Answer #112477 by Earlsdon(6294) $\"\"$ $\"About$

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1) Find the inverse of:
\n" ); document.write( " $\"f%28x%29+=+3%2Bx%5E3\"$ Rewrite this as:
\n" ); document.write( " $\"y+=+3%2Bx%5E3\"$ and exchange the x and y to get:
\n" ); document.write( " $\"x+=+3%2By%5E3\"$ Now solve this for y.
\n" ); document.write( " $\"x-3+=+y%5E3\"$ Take the cube root of both sides.
\n" ); document.write( " $\"y+=+root%283%2C%28x-3%29%29\"$
\n" ); document.write( "Now replace the y with $\"f%5E%28-1%29%28x%29\"$ to get:
\n" ); document.write( " $\"f%5E%28-1%29%28x%29+=+root%283%2C%28x-3%29%29\"$ This is the inverse of the given function.\r
\n" ); document.write( "\n" ); document.write( "2) Use Descartes' rule of signs to find the number of positive and negative real zeros.
\n" ); document.write( " $\"p%28x%29+=+2x%5E4%2B3x%5E3-2x%5E2%2Bx-2\"$ First, for the positive zeros, count the number of sign changes in the polynomial p(x):
\n" ); document.write( "There are three sign changes, so, there is a maximum of three real positive zeros, but (counting down by two's) there could also be only one real positive zero.
\n" ); document.write( "Let's look at the possible negative zeros by evaluating the function at (-x):
\n" ); document.write( " $\"p%28-x%29+=+2%28-x%29%5E4%2B3%28-x%29%5E3-2%28-x%29%5E2%2B%28-x%29-2\"$ Making the appropriate sign changes:
\n" ); document.write( " $\"p%28-x%29+=+2%28x%29%5E4-3%28x%29%5E3-2%28x%29%5E2-x-2\"$ Here, we see only one sign change, so you can expect only one negative root real zero.
\n" ); document.write( "In summary, the number of zeros (real or complex) for this function should be 4, because you have a fourth-order polynomial. However, some of those zeros may be complex.
\n" ); document.write( "In fact, if you can work it out, the function has two real zeros, one positive and one negative, and it has two complex zeros. \n" ); document.write( "

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