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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

Hi Can You Help Me To Solve Thisexercise PleaseFor Cach Polynomial LIST All POSSIBLE RATIONAL ROOTSFind All Factors Of The Leading Coefficient And Constant Valu class=

Sagot :

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:

[tex]\pm\frac{p}{q}[/tex]

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:

[tex]3x^2+2x+2[/tex]

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:

[tex]\begin{gathered} p=\pm1,\pm2 \\ q=\pm1,\pm3_{} \end{gathered}[/tex]

Hence, the possible roots are:

[tex]\begin{gathered} \frac{p}{q}=\pm\frac{1}{1},\pm\frac{1}{3},\pm\frac{2}{1},\pm\frac{2}{3}_{} \\ \therefore \\ \frac{p}{q}=\pm1,\pm\frac{1}{3},\pm2,\pm\frac{2}{3} \end{gathered}[/tex]

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