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
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asked 2021-05-08

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

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