Question

# For the following exercises, use the given information about the polynomial graph to write the equation. Degree 4. Root of multiplicity 2 at x = 4, an

Polynomial graphs

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

2021-06-21

Data: x - intercept of multiplicity $$2= 4$$
intercept of multiplicity $$1=-2,1$$
y-intercept $$= -3$$
Degree=4
Since it is a fifth degree polynomial function with multiplicity of 2 and 1 for some zeros, its general equation becomes: $$\displaystyle{f{{\left({x}\right)}}}={a}{\left({x}+{2}\right)}{\left({x}—{1}\right)}^{{2}}{\left({x}-{4}\right)}^{{2}}$$
In order to evaluate a, use the point on the graph (0,-3), therefore substitute $$f(0)=-3$$ in this equation:
$$\displaystyle-{3}={a}{\left({0}+{2}\right)}{\left({0}—{1}\right)}{\left({0}-{4}\right)}^{{2}}$$
Simplify: -$$3=-32a$$
Evaluate a: $$\displaystyle{a}=-\frac{{3}}{{-{{32}}}}=\frac{{3}}{{32}}$$
This implies that the equation of the given polynomial function is $$f(x) =$$
$$\displaystyle\frac{{3}}{{32}}{\left({x}+{2}\right)}{\left({x}—{1}\right)}{\left({x}-{4}\right)}^{{2}}$$