document.write( "Question 580756: Make up a polynomial that has the following characteristics: crosses the x-axis at -1 and 4, touches the x-axis at 0 and 2, and is above the x-axis between 0 and 2. \n" ); document.write( "

Algebra.Com's Answer #371462 by KMST(5328) $\"\"$ $\"About$

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If $\"%28x-4%29\"$ , or $\"%28x-4%29%5E3\"$ , or $\"%28x-4%29%5E3\"$ , or ... (well, you get the idea) is a factor of a polynomial, the polynomial will be zero AND change sign at x=4. Then the polynomial \"crosses the x-axis at ... 4\". To make it the simplest, we should include
\n" ); document.write( " $\"%28x%2B1%29%28x-4%29\"$ as a factor in the polynomial, so that it crosses the x-axis at -1 and 4.\"
\n" ); document.write( "On the other hand if we include $\"x%5E2\"$ and $\"%28x-2%29%5E2\"$ as factors in the polynomial, the polynomial will be zero, but will not change sign at x=0 and x=2, so we would say that it touches (but does not cross) the x-axis.
\n" ); document.write( "So $\"f%28x%29=x%5E2%28x-2%29%5E2%28x%2B1%29%28x-4%29\"$ would be a polynomial that \"crosses the x-axis at -1 and 4,\" and \"touches the x-axis at 0 and 2.\"
\n" ); document.write( "How does it behave between 0 and 2? If it is positive, then it \"is above the x-axis between 0 and 2.\"
\n" ); document.write( "But

, so we need a (-1) as an extra factor.
\n" ); document.write( " $\"P%28x%29=-x%5E2%28x-2%29%5E2%28x%2B1%29%28x-4%29\"$ is a polynomial that complies with the problem's requirements.
\n" ); document.write( " $\"graph%28300%2C300%2C-1.5%2C4.3%2C-75%2C75%2C-x%5E2%28x-2%29%5E2%28x%2B1%29%28x-4%29%29\"$ \n" ); document.write( "

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