Question 440387: `Given that f(x) =  has -1 as a zero of multiplicity 2, 2 as a zero, and -3 as a zero, find all other zeros
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! We could use synthetic division or long division and divide f(x) by a quartic polynomial, but there is a much simpler way that will take far less time.
Keep in mind that four of the zeros are known (-1, -1, 2, -3), so there are only two unknown zeros, which we will denote  and  . If you know Vieta's formulas*,
(-1)(2)(-3)r_5r_6 = -18)
This implies  and  . Applying Vieta's formulas again, if  are the roots of a quadratic polynomial, then these roots are equal to the roots of the polynomial  , which by the quadratic formula, we obtain
 upon simplification.
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*Vieta's formulas say that for any polynomial  , the sum of the roots is  and the product of the roots is  . Other cyclic sums involving the roots can be obtained, by multiplying n monomials and equating coefficients.
http://mathworld.wolfram.com/VietasFormulas.html
http://en.wikipedia.org/wiki/Vi%C3%A8te's_formulas
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