document.write( "Question 1113852: Graph the polynomial p(x)=-2x(x-3)(x+2)^3(x-2)^2 \n" ); document.write( "
Algebra.Com's Answer #728925 by greenestamps(13198)\"\" \"About 
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\n" ); document.write( "Graph the polynomial \"p%28x%29=-2x%28x-3%29%28x%2B2%29%5E3%28x-2%29%5E2\"

\n" ); document.write( "For clarification, I will write the polynomial as \"p%28x%29=%28-2%29%28x%2B2%29%5E3%28x%29%28x-2%29%5E2%28x-3%29\"

\n" ); document.write( "In this form, we can see that the polynomial is of degree 7 with a negative leading coefficient; that means the end behavior of the function is up for large negative x and down for large positive x.

\n" ); document.write( "And we can see that, from left to right, the function has
\n" ); document.write( "(1) a triple root at x=-2;
\n" ); document.write( "(2) a single root at x=0;
\n" ); document.write( "(3) a double root at x=2; and
\n" ); document.write( "(4) a single root at x=3.

\n" ); document.write( "At a single root, the function value changes sign; the graph simply crosses the x axis from positive to negative or vice versa.
\n" ); document.write( "At a double root, the sign of the function value does not change; the graph just touches the x axis, like the vertex of a parabola.
\n" ); document.write( "At a triple root, the function value changes sign, and the behavior is like the graph of x^3 -- it flattens out but then continues in the same direction.

\n" ); document.write( "So with this function, moving left to right, the graph of the function starts positive; at the triple root x=-2, it flattens out but then becomes negative; at the single root x=0, it crosses the x-axis and becomes positive; at the double root x=2 it just touches the x-axis and remains positive; and at the single root x=3 it crosses the x-axis and becomes negative.

\n" ); document.write( "Here is a graph:

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\n" ); document.write( "added at the request of the student: graph

\n" ); document.write( "\"6x%5E2%28x-4%29%5E3%28x%2B1%29%5E4\"

\n" ); document.write( "which can be written as

\n" ); document.write( "\"%286%29%28x%29%5E2%28x-4%29%5E3%28x%2B1%29%5E4\"

\n" ); document.write( "Analysis: degree 9 with positive leading coefficient, so negative for large negative x and positive for large positive x. So, \"walking\" left to right on the number line....

\n" ); document.write( "(1) negative for large negative;
\n" ); document.write( "(2) quadruple root at x=-1 (an even power), so the graph just touches the x-axis but the sign of the function does not change -- i.e., it stays negative;
\n" ); document.write( "(3) double root at x=0, so again the graph just touches the x-axis at x=0 but the function value remains negative;
\n" ); document.write( "(4) triple root at x=4 (an odd degree), so the sign of the function changes to positive; and
\n" ); document.write( "(5) there are no more zeros, so the function value remains positive the rest of the way.

\n" ); document.write( "A graph:

\n" ); document.write( "\"graph%28400%2C400%2C-2%2C6%2C-100%2C100%2C%286%29%28x%5E2%29%28x-4%29%5E3%28x%2B1%29%5E4%29\"

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