Algebra - Zeroes/Roots of Polynomials

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Section 5.2 : Zeroes/Roots of Polynomials

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

\[\begin{align*}A\left( x \right) & = {x^8} + 2{x^7} - 29{x^6} - 76{x^5} + 199{x^4} + 722{x^3} + 261{x^2} - 648x - 432\\ & = {\left( {x + 1} \right)^2}{\left( {x - 4} \right)^2}\left( {x - 1} \right){\left( {x + 3} \right)^3}\end{align*}\]

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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 : \(x = - 1\), \(x = 4\), \(x = 1\) and \(x = - 3\).

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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,

\[\begin{align*}& x = - 1:{\mbox{ multiplicity 2}}\\ & x = 4:{\mbox{ multiplicity 2}}\\ & x = 1:{\mbox{ multiplicity 1}}\\ & x = - 3:{\mbox{ multiplicity 3}}\end{align*}\]