SOLUTION: Factor The Polynomial f(x). Then solve the equation f(x)=0 1) F(x)= x^4-8x^3+17x^2+2X-24 2) -square root 7, -3i Find a polnomial function of lowest degree with rational coeffi

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Question 115792This question is from textbook Algebra For college students
: Factor The Polynomial f(x). Then solve the equation f(x)=0
1) F(x)= x^4-8x^3+17x^2+2X-24
2) -square root 7, -3i
Find a polnomial function of lowest degree with rational coefficients that has the given numbers as some zeros. I have no idea where to start
3) f(x)=x^4-12x^2-64; -2i
The polynomial function has the given zero, find the other zeros
I have no idea where to start some one point me in the right direction. This question is from textbook Algebra For college students

Answer by stanbon(75887) (Show Source):
You can put this solution on YOUR website!
Factor The Polynomial f(x). Then solve the equation f(x)=0
1) F(x)= x^4-8x^3+17x^2+2X-24
Graphing I find x=-1 is a zero. x=2 is a zero, x=3 is a zero, x=4 is a zero.
F(x) factors as (x+1)(x-2)(x-3)(x-4)
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2) -square root 7, -3i
Find a polnomial function of lowest degree with rational coefficients that has the given numbers as some zeros. I have no idea where to start
Since the coefficients are rational 3i must also be a zero.
Then: f(x) = (x-7)(x-3i)(x+3i)
f(x) = (x-7)(x^2+9)
f(x) = x^3-7x^2+9x-63
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3) f(x)=x^4-12x^2-64; -2i
The polynomial function has the given zero, find the other zeros
Since the coefficients are rational 2i must also be a zero.
So, f(x) = (x-2i)(x+2i)(?)
f(x) = (x^2+4)(?)
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Graph f(x) I find x=-4 is a zero, x=4 is a zero.
Therefore: f(x) = (x^2+4)(x-4)(x+4)
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Cheers,
Stan H.