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 and constant value of polynonnal.•ANY RATIONAL ROOTS =‡ (Constant Factor over Leading Coefficient Factor)3x^2+2x+2

Answer 1

DEFINITIONS

The Rational Roots Test (also known as Rational Zeros Theorem) allows us to find all possible rational roots of a polynomial. Suppose we have some polynomial P(x) with integer coefficients and a nonzero constant term, the possible rational roots are given in the form:

`\pm(p)/(q)`

where p represents the factors of the constant term of the polynomial and q represents the factors of the leading coefficient.

SOLUTION

The polynomial is given to be:

`3x^2+2x+2`

The leading coefficient is 3 and the constant term is 2.

Since all coefficients are integers, we can apply the rational zeros theorem.

The factors of the leading coefficient are 1 and 3, while the factors of the constant term are 1 and 2. Therefore, we have that:

`\begin{gathered} p=\pm1,\pm2 \\ q=\pm1,\pm3_{} \end{gathered}`

Hence, the possible roots are:

`\begin{gathered} (p)/(q)=\pm(1)/(1),\pm(1)/(3),\pm(2)/(1),\pm(2)/(3)_{} \\ \therefore \\ (p)/(q)=\pm1,\pm(1)/(3),\pm2,\pm(2)/(3) \end{gathered}`