Question

Find all rational zeros of f. Then use the depressed equation to find all roots of the question f(x)=0

6x^3-11x-9x-1

Answer 1

`\bf 6x^3-11x^2-9x-1=0 \qquad \begin{array}{lllrrrll} -(1)/(2)&|&6&-11&-9&-1\\ &|&&-3&7&1\\ \textendash\textendash\textendash&|&\textendash\textendash\textendash&\textendash\textendash\textendash&\textendash\textendash\textendash&\textendash\textendash\textendash\\ &&6&-14&-2&0 \end{array} \\\\ \\\\ x=-(1)/(2)\implies x+(1)/(2)=0\implies \left( x+(1)/(2) \right)(6x^2-14x-2)=0 \\\\ \textit{taking common factor on the quadratic} \\\\`
`\bf \left( x+(1)/(2) \right)2(3x^2-7x-1)=0\implies \left( x+(1)/(2) \right)(3x^2-7x-1)=0 \\\\ \textit{doing the quadratic formula on it} \\\\ x=\cfrac{-(-7)\pm√((-7)^2-4(3)(-1))}{2(3)} \\\\\\ x=\cfrac{7\pm√(49+12)}{6}\implies x=\cfrac{7\pm√(61)}{6}\impliedby \textit{61 is a prime number} \\\\\\ thus\implies x\to \begin{cases} -(1)/(2) \\\\ \cfrac{7+√(61)}{6} \\\\ \cfrac{7-√(61)}{6} \end{cases}`