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
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asked 2021-06-20

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).

Answers (1)

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 \(f(x) =\)
\(\displaystyle\frac{{3}}{{32}}{\left({x}+{2}\right)}{\left({x}—{1}\right)}{\left({x}-{4}\right)}^{{2}}\)

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