Answer:
See attached for a graph
Step-by-step explanation:
You want to know how polynomial zeros are used to create a graph, using x³ -2x² -7x -4 as an example.
Zeros
Using Descartes' rule of signs, you can determine there is one real positive zero for the given polynomial. Considering the coefficients, you can determine it is greater than 1. The rational root theorem suggests other choices are 2 and 4. You can quickly determine that x = 4 is a zero.
Dividing the polynomial by (x-4) gives x² +2x +1 = (x +1)². Then the factored form of the given polynomial is ...
= (x +1)²(x -4)
and the zeros are -1 (multiplicity 2) and +4.
Graph
The given polynomial is a cubic with a positive leading coefficient. The odd degree (3) tells you the left-end behavior will be opposite the right-end behavior. The positive leading coefficient tells you the right-end behavior will be toward (+∞, +∞). This means the general shape of the graph is from lower left to upper right.
This general shape tells you the graph will be increasing to the leftmost zero (at x=-1), and again from the rightmost zero (at x=4). The even multiplicity of the zero at x=-1 tells you the graph will touch, not cross, the x-axis there.
Immediately to the right of the zero at x=-1, the graph will be decreasing. Of course, it must have a turning point in the interval (1, 4) in order to come back up to cross the x-axis at x=4. (We know the turning point is between 1 and 4, because the graph is still decreasing at x=1 based on previous evaluations.)
At this point, you can draw the general shape of the graph as increasing to x = -1, where it touches the x-axis, then decreases to a minimum before x = 4, and increases again to cross the x-axis at x = 4 to continue to higher positive values as x increases further.
A graphing calculator plots the curve as shown in the attachment.
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Additional comment
A cubic has the interesting property that the point of inflection is halfway between any turning points. When there are turning points, they are located equidistant from the point of inflection and the point where the graph has the same y-value as the (other) turning point.
In practice, this means the minimum of the given cubic will be 1/3 of the way between the rightmost x-intercept and the "touch" point that is the local maximum. You can evaluate the polynomial at x = (4 -5/3) = 2 1/3 to find the local minimum is -500/27, about -18.52. Its coordinates are approximately (2.33, -18.52).
Polynomial graphs will generally have either a U-shape or a /-shape, depending on whether the degree is even or odd, respectively. If the leading coefficient is negative, these shapes will be inverted vertically.
The higher the multiplicity of a zero, the flatter the curve is where it meets the x-axis. This can help you judge the multiplicity of a zero from the way it meets/crosses the x-axis.
