Algebra - Zeroes/Roots of Polynomials

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Section 5.2 : Zeroes/Roots of Polynomials

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

\[f\left( x \right) = 2{x^2} + 13x - 7\]

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For this problem we’ll first need to factor the polynomial.

\[f\left( x \right) = 2{x^2} + 13x - 7 = \left( {2x - 1} \right)\left( {x + 7} \right)\]

From this we see that we have the two zeroes/roots : \(x = \frac{1}{2}\) and \(x = - 7\).

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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,

\[\begin{align*}x & = \frac{1}{2}:{\mbox{ multiplicity 1}}\\ x & = - 7:{\mbox{ multiplicity 1}}\end{align*}\]