Answer:
y = (x +1)^3(x -2)
Step-by-step explanation:
The graph is generally U-shaped and opens upward. This means the polynomial is of even degree and has a positive leading coefficient.
The x-intercepts are -1 and +2, so there will be factors of (x -(-1)) and (x -2). (An x-intercept of p corresponds to a factor (x-p).)
The curve has a flat spot at x=-1, so the multiplicity of that factor will be greater than 1. Since the axis is crossed there (not just touched), the multiplicity will be odd.
The smallest exponent of the factor (x+1) that will meet all the requirements is 3. (The sum of exponents must be even, the exponent of (x+1) must be an odd number greater than 1.)
An equation could be ...
y = (x +1)^3(x -2)
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Comment on what's to understand?
Every x-intercept of a polynomial curve corresponds to a binomial factor with a real number constant. (Above, we called the factor (x-p) for x-intercept x=p.) If the factor has odd multiplicity, the curve will cross the x-axis; if not, the curve will "touch" the x-axis at the x-intercept point. The higher the multiplicity (exponent of the factor), the "flatter" the curve is at the x-intercept.
The leading coefficient of the polynomial tells you the sign of the y-value when x takes on large positive values. If the degree of the polynomial is even, the general shape of the curve will be U-shaped, opening upward or downward depending on the sign of the leading coefficient. If the degree of the polynomial is odd, the curve will have a / or \ general shape, again depending on the sign of the leading coefficient. (\ for negative). This description is of the shape at large scales, ignoring any wiggles on the curve.
