Question

# For the following exercises, use the given information about the polynomial graph to write the equation. Degree 5. Roots of multiplicity 2 at x = −3 a

Polynomial graphs

For the following exercises, use the given information about the polynomial graph to write the equation. Degree 5. Roots of multiplicity 2 at $$x = minus;3$$ and $$x = 2$$ and a root of multiplicity 1 at $$x=minus;2$$. y-intercept at (0, 4).

2021-05-22

Data: x -ntercept of multiplicity $$2= -3,2$$
-intercept of multiplicity $$1=-2$$
y-intercept $$= 4$$
Degree$$=5$$
Since it is a fifth degree polynomial function with multiplicity of 2 and 1 for some zeros, its general equation becomes: $$\displaystyle{f{{\left({x}\right)}}}={a}{\left({x}+{3}\right)}^{{2}}{\left({x}—{2}\right)}^{{2}}{\left({x}+{2}\right)}$$
In order to evaluate a, use the point on the graph (0,4), therefore substitute $$f(0) =4$$ in this equation:
$$\displaystyle{4}={a}{\left({0}+{3}\right)}^{{2}}{\left({0}—{2}\right)}^{{2}}{\left({0}+{2}\right)}$$
Simplify: $$\displaystyle{4}={a}{\left({3}\right)}^{{2}}{\left(—{2}\right)}^{{2}}{\left({2}\right)}={72}{a}$$
Evaluate a: $$\displaystyle{a}=\frac{{4}}{{72}}=\frac{{1}}{{18}}$$
This implies that the equation of the given polynomial function is f(x) =
$$\displaystyle\frac{{1}}{{18}}{\left({x}+{3}\right)}^{{2}}{\left({x}—{2}\right)}^{{2}}{\left({x}+{2}\right)}$$