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 an

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 $$x = 3$$ and $$x = 1$$, and a root of multiplicity 1 at $$x = -3$$. y-intercept at (0, 9)

## Expert Answers (1)

2021-05-09

Data: x - intercept of multiplicity $$2=1.3$$
© - intercept of multiplicity $$1=-3$$
y-intercept $$= 9$$
Degree $$=5$$
Since it is a fifth degree polynomial function with multiplicity of 2 and 1 for some zeros, its general equation becomes: $$\displaystyle{f{{\left({x}\right)}}}={a}{\left({x}+{3}\right)}{\left({x}—{1}\right)}^{{2}}{\left({x}-{3}\right)}^{{2}}$$
In order to evaluate a, use the y - intercept (0,9), therefore substitute $$f(0)=9$$ in this equation:
$$\displaystyle{9}={a}{\left({0}+{3}\right)}{\left({0}—{1}\right)}^{{2}}{\left({0}-{3}\right)}^{{2}}$$
Simplify: $$9=27a$$
Evaluate a: $$\displaystyle{a}=\frac{{9}}{{27}}=\frac{{1}}{{3}}$$
This implies that the equation of the given polynomial function is $$f(x) =$$
$$\displaystyle\frac{{1}}{{3}}{\left({x}+{3}\right)}{\left({x}—{1}\right)}^{{2}}{\left({x}-{3}\right)}^{{2}}$$