Question

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 \((0, -4)\).

Answer 1

Data: \(x - intercept=-2,1,3\)
\(x- \text{intercept of multiplicity} =-4\)
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 \(f(x) =\)
\(\displaystyle{\left(-{\left(\frac{{2}}{{3}}\right)}\right)}{\left({x}+{3}\right)}{)}{\left({x}—{1}\right)}{\left({x}-{3}\right)}\)