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Answer:
[tex]\left(x+1\right)\left(x-2\right)\left(x+2\right)(x +2 - 4i)(x +2 + 4i)[/tex]
Step-by-step explanation:
Given
[tex]f(x) = x^5 + 5x^4 + 20x^3 - 96x - 80[/tex]
[tex]Zeros: -2, -1, 2[/tex]
Required
Factorization of f(x)
The given zeros imply that:
[tex]x=-2[/tex] [tex]x =-1[/tex] [tex]x = 2[/tex]
This gives:
[tex]x + 2 =0[/tex] [tex]x + 1 = 0[/tex] [tex]x - 2 = 0[/tex]
So, some factors are:
(x + 2), (x + 1) and (x - 2)
Divide f(x) by the factors, to get the other factors:
[tex]\frac{x^5 + 5x^4 + 20x^3 - 96x - 80}{(x + 2)(x + 1) (x - 2)}[/tex]
Using a factorization calculator, we have:
[tex]\frac{\left(x+1\right)\left(x-2\right)\left(x+2\right)\left(x^2+4x+20\right)}{(x + 2)(x + 1) (x - 2)}[/tex]
Cancel out common terms
[tex]x^2+4x+20[/tex]
Using quadratic formula, we have:
[tex]x = \frac{-b \± \sqrt{b^2 - 4ac}}{2a}[/tex]
Where
[tex]a = 1; b =4; c = 20[/tex]
[tex]x = \frac{-4 \± \sqrt{4^2 - 4*1*20}}{2*1}[/tex]
[tex]x = \frac{-4 \± \sqrt{16 - 80}}{2*1}[/tex]
[tex]x = \frac{-4 \± \sqrt{-64}}{2}[/tex]
Using complex notation
[tex]\sqrt{-64}= 8i[/tex]
So:
[tex]x = \frac{-4 \± 8i}{2}[/tex]
Simplify the fraction
[tex]x = -2 \± 4i[/tex]
Split
[tex]x = -2 + 4i \ or\ x = -2 - 4i[/tex]
Equate to 0
[tex]x +2 - 4i = 0 \ or\ x +2 + 4i = 0[/tex]
The other factors are: (x +2 - 4i) and (x +2 + 4i)
Hence, the factorization of f(x) is:
[tex]\left(x+1\right)\left(x-2\right)\left(x+2\right)(x +2 - 4i)(x +2 + 4i)[/tex]