Identify the roots of the equation. State the multiplicity of each root. 2x3 + 14x2 − 98x − 686

Find the roots of the expression `2x^3+14x^x-98x-686 ` :

We try factoring by grouping; factor out the greatest common factor of the first two terms and the last two terms:

`2x^2(x+7)-98(x+7) `

Now use the distributive property to rewrite as:

`(x+7)(2x^2-98) `

Factor the common monomial (2) from the binomial, and recognize it as the difference of two squares to get:

2(x+7)(x+7)(x-7)

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The roots are 7 and -7; -7 has multiplicity 2 and 7 has multiplicity 1

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From the graph we can see that -7 has even multiplicity as the graph touches the x-axis but does not go through:

If factoring by grouping had not worked, we would try the rational root theorem to determine if there were any rational roots.