# For the following exercises, use the given information about the polynomial graph to write the equation. Roots of multiplicity 2 at x = 3 and x = 1

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)

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