document.write( "Question 933044: How would I roughly sketch and show Y and X Intercepts: P(X)= 2x^3-x^2+x \n" ); document.write( "

Algebra.Com's Answer #566563 by josgarithmetic(39620) $\"\"$ $\"About$

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Choose the case level and stay with it.\r
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\n" ); document.write( "\n" ); document.write( " $\"p%28x%29=2x%5E3-x%5E2%2Bx=x%282x%5E2-x%2B1%29\"$ .
\n" ); document.write( "Three roots.\r
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\n" ); document.write( "\n" ); document.write( "The quadratic factor's roots:
\n" ); document.write( " $\"%281-sqrt%281-8%29%29%2F4\"$ and $\"%281%2Bsqrt%28-7%29%29%2F4\"$ , not Real. \r
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\n" ); document.write( "\n" ); document.write( "The only real root is x=0.\r
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\n" ); document.write( "\n" ); document.write( "The basic shape is that p decreases toward the left without bound and increases toward the right without bound. Between the extreme regions needs more attention.\r
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\n" ); document.write( "\n" ); document.write( "Near x=0:
\n" ); document.write( "Look at -1 and +1 to help near this.
\n" ); document.write( "At x=-1, p is 2(-)-(+)+(-), or p is (-)+(-)+(-), meaning $\"p%3C0\"$ when $\"x%3C0\"$ .
\n" ); document.write( "Look at x=1, p is 2(+)-(+)+(+), not too clear.
\n" ); document.write( "Try at x=(1/2). p=2(1/8)-1/4+1/2=1/4-1/4+1/2=1/2, which is positive.
\n" ); document.write( "The function CROSSES the x-axis at x=0.\r
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\n" ); document.write( "\n" ); document.write( "Derivative calculus would be helpful for further details.
\n" ); document.write( " $\"dp%2Fdx=6x%5E2-2x%2B1\"$ .
\n" ); document.write( "The roots of this will be the extreme points for p(x).
\n" ); document.write( "Can you handle this part of the work?\r
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\n" ); document.write( "\n" ); document.write( "Jumping ahead, this is the graph:\r
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\n" ); document.write( "\n" ); document.write( " $\"graph%28300%2C300%2C-10%2C10%2C-15%2C15%2C2x%5E3-x%5E2%2Bx%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Also, find the discriminant of the derivative. Is it negative? That tells you something.... Notice, the graph has no minimum or maximum. \n" ); document.write( "

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