SOLUTION: `Given that f(x) = {{{x^6+x^5-6x^4+4x^3+7x^2-21x-18}}} has -1 as a zero of multiplicity 2, 2 as a zero, and -3 as a zero, find all other zeros

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*,

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.

------------------
*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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