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\n" ); document.write( "I NEED HELP! :]
Algebra.Com's Answer #86024 by bucky(2189)![\"\"](\"/images/stars/stars-13.gif\")   You can put this solution on YOUR website! Let's actually make such an equation ... one that has has 3 factors that are roots with two of the \n" ); document.write( "factors being equal. We can, for example establish the equation: \n" ); document.write( ". \n" ); document.write( " \n" ); document.write( ". \n" ); document.write( "You can multiply out these factors to get the cubic equation they form. First multiply the \n" ); document.write( "two factors identical factors to get that product: \n" ); document.write( ". \n" ); document.write( " \n" ); document.write( ". \n" ); document.write( "Multiply this product by the third factor \n" ); document.write( "multiplication results in: \n" ); document.write( ". \n" ); document.write( " \n" ); document.write( ". \n" ); document.write( "The graph of this cubic equation is: \n" ); document.write( ". \n" ); document.write( " \n" ); document.write( ". \n" ); document.write( "Note what this graph tells you. The function was built using (x + 3) as one of the factors. Notice \n" ); document.write( "that this factor results in the graph crossing the x-axis at x equal to -3. The two (x – 4) factors \n" ); document.write( "cause the graph to be just tangent to the x-axis where x equals +4. So this graph illustrates a \n" ); document.write( "cubic function that has one root that is singular and has a pair of identical roots. \n" ); document.write( ". \n" ); document.write( "You can convert this function to one having three separate real roots by shifting the function \n" ); document.write( "down an appropriate amount. Suppose we start with: \n" ); document.write( ". \n" ); document.write( " \n" ); document.write( ". \n" ); document.write( "and subtract 25 from both sides. When you do that, the equation becomes: \n" ); document.write( ". \n" ); document.write( " \n" ); document.write( ". \n" ); document.write( "and the corresponding graph is: \n" ); document.write( ". \n" ); document.write( " \n" ); document.write( ". \n" ); document.write( "The graph shows now that there are three separate roots for this function. The roots are real, and \n" ); document.write( "because they are at different places on the x-axis, they are unequal roots. \n" ); document.write( ". \n" ); document.write( "For part b of this problem, the opposite is done. Instead of subtracting 25 from both sides \n" ); document.write( "of the original function, add 10 to both sides of the original function to raise the graph \n" ); document.write( "so that the point of tangency in the original graph is raised. Adding 10 to both sides results in: \n" ); document.write( ". \n" ); document.write( " \n" ); document.write( ". \n" ); document.write( "and the graph of this is the same as the original graph ... just shifted upward: \n" ); document.write( ". \n" ); document.write( " \n" ); document.write( ". \n" ); document.write( "As a result the graph crosses or touches the x-axis at one point. This means that there is \n" ); document.write( "one real root (established by the x-axis crossing) and two complex roots (having an imaginary parts) \n" ); document.write( "to comprise the three roots of the cubic function. \n" ); document.write( ". \n" ); document.write( "Hope that this helps you to understand what the problem was asking for, how to do it, and \n" ); document.write( "what the results mean. \n" ); document.write( ". \n" ); document.write( " |