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
factors being equal. We can, for example establish the equation:
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You can multiply out these factors to get the cubic equation they form. First multiply the
two factors identical factors to get that product:
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Multiply this product by the third factor . Without going through all the work, this
multiplication results in:
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The graph of this cubic equation is:
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Note what this graph tells you. The function was built using (x + 3) as one of the factors. Notice
that this factor results in the graph crossing the x-axis at x equal to -3. The two (x – 4) factors
cause the graph to be just tangent to the x-axis where x equals +4. So this graph illustrates a
cubic function that has one root that is singular and has a pair of identical roots.
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You can convert this function to one having three separate real roots by shifting the function
down an appropriate amount. Suppose we start with:
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and subtract 25 from both sides. When you do that, the equation becomes:
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and the corresponding graph is:
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The graph shows now that there are three separate roots for this function. The roots are real, and
because they are at different places on the x-axis, they are unequal roots.
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For part b of this problem, the opposite is done. Instead of subtracting 25 from both sides
of the original function, add 10 to both sides of the original function to raise the graph
so that the point of tangency in the original graph is raised. Adding 10 to both sides results in:
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and the graph of this is the same as the original graph ... just shifted upward:
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As a result the graph crosses or touches the x-axis at one point. This means that there is
one real root (established by the x-axis crossing) and two complex roots (having an imaginary parts)
to comprise the three roots of the cubic function.
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Hope that this helps you to understand what the problem was asking for, how to do it, and
what the results mean.
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