document.write( "Question 1100994: What does the degree of a polynomial expression tell you about its related polynomial function? Explain your thinking. Give an example of a polynomial expression of degree three. Provide information regarding the graph and zeros of the related polynomial function. \n" ); document.write( "

Algebra.Com's Answer #715602 by KMST(5328) $\"\"$ $\"About$

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WHAT COULD YOU BE EXPECTED TO BE THINKING?
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\n" ); document.write( "Maybe you are expected to think like a calculus/pre-calculus student:
\n" ); document.write( "The end behavior of a polynomial function of even degree
\n" ); document.write( "can be represented by this

.
\n" ); document.write( "The graph of function like that may may never cross the x-axis,
\n" ); document.write( "so the function could have no real zeros.
\n" ); document.write( "If it does have zeros, they will come in pairs,
\n" ); document.write( "because \"what goes up must come down\" and vice versa.
\n" ); document.write( "Of course the function cannot have more zeros than its degree.
\n" ); document.write( "For example, a polynomial function of degree 6 could have 0, 2, 4, or 6 real zeros.
\n" ); document.write( "
\n" ); document.write( "The end behavior of a polynomial function of even degree
\n" ); document.write( "can be represented by this

.
\n" ); document.write( "The graph of function like that must cross the x-axis at least once,
\n" ); document.write( "so the function must have at least one real zero.
\n" ); document.write( "There and be up to as many real zeros as the degree of the polynomial function , and there will be an odd number of real zeros,
\n" ); document.write( "because an even number would mean an end-behavior like the one described by even degree polynomial functions.
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\n" ); document.write( "Maybe you are expected to think in terms of polynomial factoring:
\n" ); document.write( "A polynomial of degree $\"n\"$ must have $\"n\"$ complex number zeros,
\n" ); document.write( "and could be written as
\n" ); document.write( " $\"P%28x%29=a%28x-z%5B1%5D%29%2A%28x-z%5B2%5D%29%2A%22...%22%2A%28x-z%5Bn%5D%29\"$ , where $\"a\"$ is the leading coefficient.
\n" ); document.write( "Some of those complex number zeros may not be real numbers,
\n" ); document.write( "but those will come as $\"p\"$ pairs of conjugate complex number zeros,
\n" ); document.write( "and in that case, a polynomial function with real coefficients can be written as
\n" ); document.write( " $\"P%28x%29=a%28x-r%5B1%5D%29%2A%28x-r%5B2%5D%29%2A%22...%22%2A%28x-r%5Bn-2p%5D%29%2AQ%28x%29\"$ where $\"Q%28x%29\"$ is a polynomial of degree $\"2p%3C=n\"$ with no real zeros.
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\n" ); document.write( "An example of a polynomial function of degree 3 is
\n" ); document.write( " $\"f%28x%29=x%5E3-5x=x%28x%5E2-5%29=%28x-sqrt%285%29%29%28x%2Bsqrt%285%29%29\"$
\n" ); document.write( "It has 3 real zeros: $\"0\"$ , $\"-sqrt%285%29\"$ and $\"sqrt%285%29\"$ .
\n" ); document.write( "For $\"x%3Esqrt%285%29\"$ all 3 factors are positive and so is the function.
\n" ); document.write( "Between $\"0\"$ and $\"sqrt%285%29\"$ , $\"%28x-sqrt%285%29%29\"$ is negative,
\n" ); document.write( "but the other two factors are positive, so the function is negative.
\n" ); document.write( "Between $\"sqrt%285%29\"$ and $\"0\"$ (for $\"-sqrt%285%29%3Cx%3C0\"$ ),
\n" ); document.write( "two factors are negative , $\"x%3C0\"$ and $\"%28x-sqrt%285%29%29%3C0\"$ ,
\n" ); document.write( "but the other factor is positive, $\"%28x%2Bsqrt%285%29%29%3E0\"$ , and the function is positive.
\n" ); document.write( "For $\"x%3C-sqrt%285%29\"$ , all three factors are negative, and so is the function.
\n" ); document.write( " $\"graph%28300%2C300%2C-3%2C3%2C-4.5%2C4.5%2C-5%2Cx%5E3-5x%29\"$ \n" ); document.write( "

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