Algebra - Zeroes/Roots of Polynomials

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Section 5.2 : Zeroes/Roots of Polynomials

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2. List all of the zeros of the following polynomial and give their multiplicities.

\[g\left( x \right) = {x^6} - 3{x^5} - 6{x^4} + 10{x^3} + 21{x^2} + 9x = x{\left( {x - 3} \right)^2}{\left( {x + 1} \right)^3}\]

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Start Solution

For this problem the polynomial has already been factored and so all we need to do is get the zeroes/roots from the factored form.

The zeroes/roots of this polynomial are : \(x = 0\), \(x = 3\) and \(x = - 1\).

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For the multiplicities just remember that the multiplicity of the zero/root is simply the exponent on the term that produces the zero/root. Therefore, the multiplicities of each zero/root is,

\[\begin{align*}& x = 0:{\mbox{ multiplicity 1}}\\ & x = 3:{\mbox{ multiplicity 2}}\\ & x = - 1:{\mbox{ multiplicity 3}}\end{align*}\]