Answer:
a) [tex]P (x) = (x + 3) (x-1) (x-4)[/tex]
b) [tex]P (x) = (2x + 5) (5x - 4) (x-6)[/tex]
c) [tex]P (x) = (x-3) (x-1) (x-4) (x + 1) ^ 2[/tex]
Step-by-step explanation:
For the question a * you need to find a polynomial of degree 3 with zeros in -3, 1 and 4.
This means that the polynomial P(x) must be zero when x = -3, x = 1 and x = 4.
Then write the polynomial in factored form.
[tex]P (x) = (x + 3) (x-1) (x-4)[/tex]
Note that this polynomial has degree 3 and is zero at x = -3, x = 1 and x = 4.
For question b, do the same procedure.
Degree: 3
Zeros: -5/2, 4/5, 6.
The factors are
[tex]x = -\frac{5}{2}\\\\x +\frac{5}{2} = 0\\\\(2x +5) = 0[/tex]
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[tex]x =\frac{4}{5}\\\\x-\frac{4}{5} = 0\\\\(5x-4) = 0[/tex]
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[tex]x = 6\\\\(x-6) = 0[/tex]
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[tex]P (x) = (2x + 5) (5x - 4) (x-6)[/tex]
Finally for the question c we have
Degree: 5
Zeros: -3, 1, 4, -1
Multiplicity 2 in -1
[tex]x = -3\\\\(x-3) = 0[/tex]
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[tex]x = 1\\\\(x-1) = 0[/tex]
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[tex]x = 4\\\\(x-4) = 0[/tex]
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[tex]x = -1\\\\(x + 1) = 0[/tex]
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[tex]P (x) = (x-3) (x-1) (x-4) (x + 1) ^ 2[/tex]
