Question

For the equation 2x 3 - 11x 2 + 12x + 9 = 0,the root 3 has a multiplicity of _____.

Answer 1

2x^3 - 11x^2 + 12x + 9
= (x - 3)^2( 2x = 1)

The root 3 has multiplicity of 2.

Answer 2

The multiplicity of 3 is 2

Explanation:

`Use\:the\:rational\:root\:theorem`

`a_0=9,\:\quad a_n=2`

`\mathrm{The\:dividers\:of\:}a_0:\quad 1,\:3,\:9,\:\quad \mathrm{The\:dividers\:of\:}a_n:\quad 1,\:2`

`\mathrm{Therefore,\:check\:the\:following\:rational\:numbers:\quad }\pm (1,\:3,\:9)/(1,\:2)`

`(3)/(1)\mathrm{\:is\:a\:root\:of\:the\:expression,\:so\:factor\:out\:}x-3`

`=\left(x-3\right)(2x^3-11x^2+12x+9)/(x-3)`

since,
`(2x^3-11x^2+12x+9)/(x-3)=2x^2-5x-3`

`\mathrm{Factor}\:2x^2-5x-3:\quad \left(2x+1\right)\left(x-3\right)`

`=\left(x-3\right)\left(2x+1\right)\left(x-3\right)`

`=\left(x-3\right)^2\left(2x+1\right)`

Therefore, the multiplicity of 3 is 2