SOLUTION: Find a cubic polynomial w/ integer coefficients & roots: positive 1, -1/4, and positive 3/2.

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Question 1040832: Find a cubic polynomial w/ integer coefficients & roots: positive 1, -1/4, and positive 3/2.
Found 3 solutions by Boreal, solver91311, ikleyn:
Answer by Boreal(15235)   (Show Source): You can put this solution on YOUR website!
The factors would be (x-1)(4x+1)(2x-3). Set each of those equal to zero, and the roots 1, -1/4, and 3/2 would occur.
Foil the second and third and get (8x^2-10x-3). Multiply that by (x-1) for 8x^3-8x^2-10x^2+10x-3x+3
The polynomial is 8x^3-18x^2+7x+3.
Answer by solver91311(24713)   (Show Source): You can put this solution on YOUR website!


If is a zero of a polynomial, then is a factor of the polynomial. You have three zeros given, so just create the factors that are associated with each one and multiply them together. The rational zeros make it a little trickier; just note that a factor of is the same thing as .

John

My calculator said it, I believe it, that settles it
Answer by ikleyn(52858)   (Show Source): You can put this solution on YOUR website!
.
Find a cubic polynomial w/ integer coefficients & roots: positive 1, -1/4, and positive 3/2.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

What John said is an algorithm, i.e. a "software".
To transform it into a "hardware", consider first the polynomial

p(x) =  = .

It has the assigned roots, obviously.
When you open the parentheses, you will get the polynomial with rational coefficients, not with integer yet.

In order to get the polynomial with integer coefficients, mulptiply p(x) by the denominators of fractions 4*2. You will get

f(x) = 4*2*p(x) = .

This polynomial is exactly what you need. 
It has the given values as the roots and integer coefficients.

Notice that what I said is consistent with the solutions of the two other tutors.


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