For the following exercises, use the given information about the polynomial graph to write the equation. Zeros at x = −2, x = 1, and x = 3. (0, −4).

For the following exercises, use the given information about the polynomial graph to write the equation. Degree 3. Zeros at x = −2, x = 1, and x = 3. y-intercept at (0, −4).

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Ezra Herbert
Data: x — intercept=-2,1,3
© — intercept of multiplicity =-4
Degree=4
Since it is a third degree function with three x intercepts, its general equation becomes: $$\displaystyle{f{{\left({x}\right)}}}={a}{\left({x}+{2}\right)}{\left({x}—{1}\right)}{\left({x}-{3}\right)}$$
In order to evaluate a, use the y - intercept (0,-4), therefore substitute f(0)=-4 in this equation:
$$\displaystyle-{4}={a}{\left({0}+{2}\right)}{\left({0}—{1}\right)}{\left({0}-{3}\right)}$$
Simplify: -4=6a
Evaluate a: $$\displaystyle{a}=-\frac{{4}}{{6}}=-{\left(\frac{{2}}{{3}}\right)}$$
This implies that the equation of the given polynomial function is f(x) =
$$\displaystyle{\left(-{\left(\frac{{2}}{{3}}\right)}\right)}{\left({x}+{3}\right)}{)}{\left({x}—{1}\right)}{\left({x}-{3}\right)}$$