Question

Hi, can you help me to solve thisexercise, please!!For cach polynomial, LIST all POSSIBLE RATIONAL ROOTS•Find all factors of the leading coefficient andconstant value of polynonnal.•ANY RATIONAL ROOTS =‡ (Constant Factor over Leading Coefficient Factor)6x^3+7x^2-3x-1

Answer 1

`\begin{gathered} Possible\: Roots\colon\pm1,\pm(1)/(2),\pm(1)/(3),\pm(1)/(6) \\ Actual\: Rational\: Roots\colon\: None \end{gathered}`

1) We can do this by listing all the factors of -1, and the leading coefficient 6. So, we can write them as a ratio this way:

`(p)/(q)=\pm(1)/(1,\:2,\:3,\:6)`

Note that p stands for the constant and q the factors of that leading coefficient

2) Now, let's test them by plugging them into the polynomial. If it is a rational root it must yield zero:

`\begin{gathered} 6x^3+7x^2-3x+1=0 \\ 6(\pm1)^3+7(\pm1)^2-3(\pm1)+1=0 \\ 71\\e0,5\\e0 \\ (1)/(2),-(1)/(2) \\ 6(\pm(1)/(2))^3+7(\pm(1)/(2))^2-3(\pm(1)/(2))+1=0 \\ 2\\e0,(7)/(2)\\e0 \\ \\ 6(\pm(1)/(3))^3+7(\pm(1)/(3))^2-3(\pm(1)/(3))+1=0 \\ 1\\e0,(23)/(9)\\e0 \\ (1)/(6),-(1)/(6) \\ 6((1)/(6))^3+7((1)/(6))^2-3((1)/(6))+1=0 \\ (13)/(18)\\e0,-(5)/(3)\\e0 \end{gathered}`

3) So the possible roots are:

`\pm1,\pm(1)/(2),\pm(1)/(3),\pm(1)/(6)`

But there are no actual rational roots.