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.

Polynomial graphs

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 $$(0, -4)$$.

2021-06-09

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 $$f(x) =$$
$$\displaystyle{\left(-{\left(\frac{{2}}{{3}}\right)}\right)}{\left({x}+{3}\right)}{)}{\left({x}—{1}\right)}{\left({x}-{3}\right)}$$