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 a

Polynomial graphs
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asked 2021-05-21

For the following exercises, use the given information about the polynomial graph to write the equation. Degree 5. Roots of multiplicity 2 at \(x = minus;3 \) and \(x = 2\) and a root of multiplicity 1 at \(x=minus;2\). y-intercept at (0, 4).

Answers (1)

2021-05-22

Data: x -ntercept of multiplicity \(2= -3,2\)
-intercept of multiplicity \(1=-2\)
y-intercept \(= 4\)
Degree\(=5\)
Since it is a fifth degree polynomial function with multiplicity of 2 and 1 for some zeros, its general equation becomes: \(\displaystyle{f{{\left({x}\right)}}}={a}{\left({x}+{3}\right)}^{{2}}{\left({x}—{2}\right)}^{{2}}{\left({x}+{2}\right)}\)
In order to evaluate a, use the point on the graph (0,4), therefore substitute \(f(0) =4\) in this equation:
\(\displaystyle{4}={a}{\left({0}+{3}\right)}^{{2}}{\left({0}—{2}\right)}^{{2}}{\left({0}+{2}\right)}\)
Simplify: \(\displaystyle{4}={a}{\left({3}\right)}^{{2}}{\left(—{2}\right)}^{{2}}{\left({2}\right)}={72}{a}\)
Evaluate a: \(\displaystyle{a}=\frac{{4}}{{72}}=\frac{{1}}{{18}}\)
This implies that the equation of the given polynomial function is f(x) =
\(\displaystyle\frac{{1}}{{18}}{\left({x}+{3}\right)}^{{2}}{\left({x}—{2}\right)}^{{2}}{\left({x}+{2}\right)}\)

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